﻿ Trigonometry: Multiple-Angle Formulas

# Trigonometry: Multiple-Angle Formulas

## Definition and Motivation

In trigonometry, the multiple-angle formulas make it possible to decompose $\sin(na)$ and $\cos(na)$ into polynomials in $\sin(a)$ and $\cos(a)$. They can be obtained by computing the Chebyshev polynomials of first and second kind. See:

The Chebyshev polynomials of first kind are defined by the following recurrence: $\quad\\ \quad\begin{cases} \forall x \in [-1, 1], \ \ T_{0}(x) = 1, \ T_{1}(x) = x\\[2mm] \forall n \in \mathbb{N} \setminus \{0, 1\}, \ \forall x \in [-1, 1], \ \ T_{n+1}(x) = 2x \cdot T_{n}(x) - T_{n-1}(x) \end{cases}$

The Chebyshev polynomials of second kind are defined by the following recurrence: $\quad\\ \quad\begin{cases} \forall x \in [-1, 1], \ \ U_{0}(x) = 1, \ U_{1}(x) = 2x\\[2mm] \forall n \in \mathbb{N} \setminus \{0, 1\}, \ \forall x \in [-1, 1], \ \ U_{n+1}(x) = 2x \cdot U_{n}(x) - U_{n-1}(x) \end{cases}$

The multiple-angle formulas for $\sin(na)$ and $\cos(na)$ are defined as follows: $\quad\\ \quad\begin{cases} \forall n \in \mathbb{N} \setminus \{0\}, \ \ \sin(2na) = (-1)^{n+1} \cdot U_{2n-1}\big(\sin(a)\big) \cdot \cos(a)\\[2mm] \forall n \in \mathbb{N} \cup \{0\}, \ \ \sin\big((2n+1)a\big) = (-1)^{n} \cdot T_{2n+1}\big(\sin(a)\big)\\[2mm] \forall n \in \mathbb{N} \cup \{0\}, \ \ \cos(na) = T_{n}\big(\cos(a)\big) \end{cases}$

The multiple-angle formulas presented below were calculated using a Python program (with infinite-precision integers) that I wrote in 2012.

The coefficients for $n = 0 \mathinner{\ldotp\ldotp} 100$ were checked against the sequence A039991 of the Online Encyclopedia of Integer Sequences (OEIS).

## Decomposition of $\sin(na)$ as a sum of multiples of $\sin^{k}(a), \ k \le n,$ for $n = 1 \mathinner{\ldotp\ldotp} 100$

 $\sin(a) = \, \sin(a)$ $\sin(2a) = 2\, \sin(a)\ \cos(a)$ $\sin(3a) = - 4\, \sin^{3}(a) + 3\, \sin(a)$ $\sin(4a) = \big( - 8\, \sin^{3}(a) + 4\, \sin(a)\big) \cdot \cos(a)$ $\sin(5a) = 16\, \sin^{5}(a) - 20\, \sin^{3}(a) + 5\, \sin(a)$ $\sin(6a) = \big(32\, \sin^{5}(a) - 32\, \sin^{3}(a) + 6\, \sin(a)\big) \cdot \cos(a)$ $\sin(7a) = - 64\, \sin^{7}(a) + 112\, \sin^{5}(a) - 56\, \sin^{3}(a) + 7\, \sin(a)$ $\sin(8a) = \big( - 128\, \sin^{7}(a) + 192\, \sin^{5}(a) - 80\, \sin^{3}(a) + 8\, \sin(a)\big) \cdot \cos(a)$ $\sin(9a) = 256\, \sin^{9}(a) - 576\, \sin^{7}(a) + 432\, \sin^{5}(a) - 120\, \sin^{3}(a) + 9\, \sin(a)$ $\sin(10a) = \big(512\, \sin^{9}(a) - 1024\, \sin^{7}(a) + 672\, \sin^{5}(a) - 160\, \sin^{3}(a) + 10\, \sin(a)\big) \cdot \cos(a)$ $\sin(11a) = - 1024\, \sin^{11}(a) + 2816\, \sin^{9}(a) - 2816\, \sin^{7}(a) + 1232\, \sin^{5}(a) - 220\, \sin^{3}(a) + 11\, \sin(a)$ $\sin(12a) = \big( - 2048\, \sin^{11}(a) + 5120\, \sin^{9}(a) - 4608\, \sin^{7}(a) + 1792\, \sin^{5}(a) - 280\, \sin^{3}(a) + 12\, \sin(a)\big) \cdot \cos(a)$ $\sin(13a) = 4096\, \sin^{13}(a) - 13312\, \sin^{11}(a) + 16640\, \sin^{9}(a) - 9984\, \sin^{7}(a) + 2912\, \sin^{5}(a) - 364\, \sin^{3}(a) + 13\, \sin(a)$ $\sin(14a) = \big(8192\, \sin^{13}(a) - 24576\, \sin^{11}(a) + 28160\, \sin^{9}(a) - 15360\, \sin^{7}(a) + 4032\, \sin^{5}(a) - 448\, \sin^{3}(a) + 14\, \sin(a)\big) \cdot \cos(a)$ $\sin(15a) = - 16384\, \sin^{15}(a) + 61440\, \sin^{13}(a) - 92160\, \sin^{11}(a) + 70400\, \sin^{9}(a) - 28800\, \sin^{7}(a) + 6048\, \sin^{5}(a) - 560\, \sin^{3}(a) + 15\, \sin(a)$ $\sin(16a) = \big( - 32768\, \sin^{15}(a) + 114688\, \sin^{13}(a) - 159744\, \sin^{11}(a) + 112640\, \sin^{9}(a) - 42240\, \sin^{7}(a) + 8064\, \sin^{5}(a) - 672\, \sin^{3}(a) + 16\, \sin(a)\big) \cdot \cos(a)$ $\sin(17a) = 65536\, \sin^{17}(a) - 278528\, \sin^{15}(a) + 487424\, \sin^{13}(a) - 452608\, \sin^{11}(a) + 239360\, \sin^{9}(a) - 71808\, \sin^{7}(a) + 11424\, \sin^{5}(a) - 816\, \sin^{3}(a) + 17\, \sin(a)$ $\sin(18a) = \big(131072\, \sin^{17}(a) - 524288\, \sin^{15}(a) + 860160\, \sin^{13}(a) - 745472\, \sin^{11}(a) + 366080\, \sin^{9}(a) - 101376\, \sin^{7}(a) + 14784\, \sin^{5}(a) - 960\, \sin^{3}(a) + 18\, \sin(a)\big) \cdot \cos(a)$ $\sin(19a) = - 262144\, \sin^{19}(a) + 1245184\, \sin^{17}(a) - 2490368\, \sin^{15}(a) + 2723840\, \sin^{13}(a) - 1770496\, \sin^{11}(a) + 695552\, \sin^{9}(a) - 160512\, \sin^{7}(a) + 20064\, \sin^{5}(a) - 1140\, \sin^{3}(a) + 19\, \sin(a)$ $\sin(20a) = \big( - 524288\, \sin^{19}(a) + 2359296\, \sin^{17}(a) - 4456448\, \sin^{15}(a) + 4587520\, \sin^{13}(a) - 2795520\, \sin^{11}(a) + 1025024\, \sin^{9}(a) - 219648\, \sin^{7}(a) + 25344\, \sin^{5}(a) - 1320\, \sin^{3}(a) + 20\, \sin(a)\big) \cdot \cos(a)$ $\sin(21a) = 1048576\, \sin^{21}(a) - 5505024\, \sin^{19}(a) + 12386304\, \sin^{17}(a) - 15597568\, \sin^{15}(a) + 12042240\, \sin^{13}(a) - 5870592\, \sin^{11}(a) + 1793792\, \sin^{9}(a) - 329472\, \sin^{7}(a) + 33264\, \sin^{5}(a) - 1540\, \sin^{3}(a) + 21\, \sin(a)$ $\sin(22a) = \big(2097152\, \sin^{21}(a) - 10485760\, \sin^{19}(a) + 22413312\, \sin^{17}(a) - 26738688\, \sin^{15}(a) + 19496960\, \sin^{13}(a) - 8945664\, \sin^{11}(a) + 2562560\, \sin^{9}(a) - 439296\, \sin^{7}(a) + 41184\, \sin^{5}(a) - 1760\, \sin^{3}(a) + 22\, \sin(a)\big) \cdot \cos(a)$ $\sin(23a) = - 4194304\, \sin^{23}(a) + 24117248\, \sin^{21}(a) - 60293120\, \sin^{19}(a) + 85917696\, \sin^{17}(a) - 76873728\, \sin^{15}(a) + 44843008\, \sin^{13}(a) - 17145856\, \sin^{11}(a) + 4209920\, \sin^{9}(a) - 631488\, \sin^{7}(a) + 52624\, \sin^{5}(a) - 2024\, \sin^{3}(a) + 23\, \sin(a)$ $\sin(24a) = \big( - 8388608\, \sin^{23}(a) + 46137344\, \sin^{21}(a) - 110100480\, \sin^{19}(a) + 149422080\, \sin^{17}(a) - 127008768\, \sin^{15}(a) + 70189056\, \sin^{13}(a) - 25346048\, \sin^{11}(a) + 5857280\, \sin^{9}(a) - 823680\, \sin^{7}(a) + 64064\, \sin^{5}(a) - 2288\, \sin^{3}(a) + 24\, \sin(a)\big) \cdot \cos(a)$ $\sin(25a) = 16777216\, \sin^{25}(a) - 104857600\, \sin^{23}(a) + 288358400\, \sin^{21}(a) - 458752000\, \sin^{19}(a) + 466944000\, \sin^{17}(a) - 317521920\, \sin^{15}(a) + 146227200\, \sin^{13}(a) - 45260800\, \sin^{11}(a) + 9152000\, \sin^{9}(a) - 1144000\, \sin^{7}(a) + 80080\, \sin^{5}(a) - 2600\, \sin^{3}(a) + 25\, \sin(a)$ $\sin(26a) = \big(33554432\, \sin^{25}(a) - 201326592\, \sin^{23}(a) + 530579456\, \sin^{21}(a) - 807403520\, \sin^{19}(a) + 784465920\, \sin^{17}(a) - 508035072\, \sin^{15}(a) + 222265344\, \sin^{13}(a) - 65175552\, \sin^{11}(a) + 12446720\, \sin^{9}(a) - 1464320\, \sin^{7}(a) + 96096\, \sin^{5}(a) - 2912\, \sin^{3}(a) + 26\, \sin(a)\big) \cdot \cos(a)$ $\sin(27a) = - 67108864\, \sin^{27}(a) + 452984832\, \sin^{25}(a) - 1358954496\, \sin^{23}(a) + 2387607552\, \sin^{21}(a) - 2724986880\, \sin^{19}(a) + 2118057984\, \sin^{17}(a) - 1143078912\, \sin^{15}(a) + 428654592\, \sin^{13}(a) - 109983744\, \sin^{11}(a) + 18670080\, \sin^{9}(a) - 1976832\, \sin^{7}(a) + 117936\, \sin^{5}(a) - 3276\, \sin^{3}(a) + 27\, \sin(a)$ $\sin(28a) = \big( - 134217728\, \sin^{27}(a) + 872415232\, \sin^{25}(a) - 2516582400\, \sin^{23}(a) + 4244635648\, \sin^{21}(a) - 4642570240\, \sin^{19}(a) + 3451650048\, \sin^{17}(a) - 1778122752\, \sin^{15}(a) + 635043840\, \sin^{13}(a) - 154791936\, \sin^{11}(a) + 24893440\, \sin^{9}(a) - 2489344\, \sin^{7}(a) + 139776\, \sin^{5}(a) - 3640\, \sin^{3}(a) + 28\, \sin(a)\big) \cdot \cos(a)$ $\sin(29a) = 268435456\, \sin^{29}(a) - 1946157056\, \sin^{27}(a) + 6325010432\, \sin^{25}(a) - 12163481600\, \sin^{23}(a) + 15386804224\, \sin^{21}(a) - 13463453696\, \sin^{19}(a) + 8341487616\, \sin^{17}(a) - 3683254272\, \sin^{15}(a) + 1151016960\, \sin^{13}(a) - 249387008\, \sin^{11}(a) + 36095488\, \sin^{9}(a) - 3281408\, \sin^{7}(a) + 168896\, \sin^{5}(a) - 4060\, \sin^{3}(a) + 29\, \sin(a)$ $\sin(30a) = \big(536870912\, \sin^{29}(a) - 3758096384\, \sin^{27}(a) + 11777605632\, \sin^{25}(a) - 21810380800\, \sin^{23}(a) + 26528972800\, \sin^{21}(a) - 22284337152\, \sin^{19}(a) + 13231325184\, \sin^{17}(a) - 5588385792\, \sin^{15}(a) + 1666990080\, \sin^{13}(a) - 343982080\, \sin^{11}(a) + 47297536\, \sin^{9}(a) - 4073472\, \sin^{7}(a) + 198016\, \sin^{5}(a) - 4480\, \sin^{3}(a) + 30\, \sin(a)\big) \cdot \cos(a)$ $\sin(31a) = - 1073741824\, \sin^{31}(a) + 8321499136\, \sin^{29}(a) - 29125246976\, \sin^{27}(a) + 60850962432\, \sin^{25}(a) - 84515225600\, \sin^{23}(a) + 82239815680\, \sin^{21}(a) - 57567870976\, \sin^{19}(a) + 29297934336\, \sin^{17}(a) - 10827497472\, \sin^{15}(a) + 2870927360\, \sin^{13}(a) - 533172224\, \sin^{11}(a) + 66646528\, \sin^{9}(a) - 5261568\, \sin^{7}(a) + 236096\, \sin^{5}(a) - 4960\, \sin^{3}(a) + 31\, \sin(a)$ $\sin(32a) = \big( - 2147483648\, \sin^{31}(a) + 16106127360\, \sin^{29}(a) - 54492397568\, \sin^{27}(a) + 109924319232\, \sin^{25}(a) - 147220070400\, \sin^{23}(a) + 137950658560\, \sin^{21}(a) - 92851404800\, \sin^{19}(a) + 45364543488\, \sin^{17}(a) - 16066609152\, \sin^{15}(a) + 4074864640\, \sin^{13}(a) - 722362368\, \sin^{11}(a) + 85995520\, \sin^{9}(a) - 6449664\, \sin^{7}(a) + 274176\, \sin^{5}(a) - 5440\, \sin^{3}(a) + 32\, \sin(a)\big) \cdot \cos(a)$ $\sin(33a) = 4294967296\, \sin^{33}(a) - 35433480192\, \sin^{31}(a) + 132875550720\, \sin^{29}(a) - 299708186624\, \sin^{27}(a) + 453437816832\, \sin^{25}(a) - 485826232320\, \sin^{23}(a) + 379364311040\, \sin^{21}(a) - 218864025600\, \sin^{19}(a) + 93564370944\, \sin^{17}(a) - 29455450112\, \sin^{15}(a) + 6723526656\, \sin^{13}(a) - 1083543552\, \sin^{11}(a) + 118243840\, \sin^{9}(a) - 8186112\, \sin^{7}(a) + 323136\, \sin^{5}(a) - 5984\, \sin^{3}(a) + 33\, \sin(a)$ $\sin(34a) = \big(8589934592\, \sin^{33}(a) - 68719476736\, \sin^{31}(a) + 249644974080\, \sin^{29}(a) - 544923975680\, \sin^{27}(a) + 796951314432\, \sin^{25}(a) - 824432394240\, \sin^{23}(a) + 620777963520\, \sin^{21}(a) - 344876646400\, \sin^{19}(a) + 141764198400\, \sin^{17}(a) - 42844291072\, \sin^{15}(a) + 9372188672\, \sin^{13}(a) - 1444724736\, \sin^{11}(a) + 150492160\, \sin^{9}(a) - 9922560\, \sin^{7}(a) + 372096\, \sin^{5}(a) - 6528\, \sin^{3}(a) + 34\, \sin(a)\big) \cdot \cos(a)$ $\sin(35a) = - 17179869184\, \sin^{35}(a) + 150323855360\, \sin^{33}(a) - 601295421440\, \sin^{31}(a) + 1456262348800\, \sin^{29}(a) - 2384042393600\, \sin^{27}(a) + 2789329600512\, \sin^{25}(a) - 2404594483200\, \sin^{23}(a) + 1551944908800\, \sin^{21}(a) - 754417664000\, \sin^{19}(a) + 275652608000\, \sin^{17}(a) - 74977509376\, \sin^{15}(a) + 14910300160\, \sin^{13}(a) - 2106890240\, \sin^{11}(a) + 202585600\, \sin^{9}(a) - 12403200\, \sin^{7}(a) + 434112\, \sin^{5}(a) - 7140\, \sin^{3}(a) + 35\, \sin(a)$ $\sin(36a) = \big( - 34359738368\, \sin^{35}(a) + 292057776128\, \sin^{33}(a) - 1133871366144\, \sin^{31}(a) + 2662879723520\, \sin^{29}(a) - 4223160811520\, \sin^{27}(a) + 4781707886592\, \sin^{25}(a) - 3984756572160\, \sin^{23}(a) + 2483111854080\, \sin^{21}(a) - 1163958681600\, \sin^{19}(a) + 409541017600\, \sin^{17}(a) - 107110727680\, \sin^{15}(a) + 20448411648\, \sin^{13}(a) - 2769055744\, \sin^{11}(a) + 254679040\, \sin^{9}(a) - 14883840\, \sin^{7}(a) + 496128\, \sin^{5}(a) - 7752\, \sin^{3}(a) + 36\, \sin(a)\big) \cdot \cos(a)$ $\sin(37a) = 68719476736\, \sin^{37}(a) - 635655159808\, \sin^{35}(a) + 2701534429184\, \sin^{33}(a) - 6992206757888\, \sin^{31}(a) + 12315818721280\, \sin^{29}(a) - 15625695002624\, \sin^{27}(a) + 14743599316992\, \sin^{25}(a) - 10531142369280\, \sin^{23}(a) + 5742196162560\, \sin^{21}(a) - 2392581734400\, \sin^{19}(a) + 757650882560\, \sin^{17}(a) - 180140769280\, \sin^{15}(a) + 31524634624\, \sin^{13}(a) - 3940579328\, \sin^{11}(a) + 336540160\, \sin^{9}(a) - 18356736\, \sin^{7}(a) + 573648\, \sin^{5}(a) - 8436\, \sin^{3}(a) + 37\, \sin(a)$ $\sin(38a) = \big(137438953472\, \sin^{37}(a) - 1236950581248\, \sin^{35}(a) + 5111011082240\, \sin^{33}(a) - 12850542149632\, \sin^{31}(a) + 21968757719040\, \sin^{29}(a) - 27028229193728\, \sin^{27}(a) + 24705490747392\, \sin^{25}(a) - 17077528166400\, \sin^{23}(a) + 9001280471040\, \sin^{21}(a) - 3621204787200\, \sin^{19}(a) + 1105760747520\, \sin^{17}(a) - 253170810880\, \sin^{15}(a) + 42600857600\, \sin^{13}(a) - 5112102912\, \sin^{11}(a) + 418401280\, \sin^{9}(a) - 21829632\, \sin^{7}(a) + 651168\, \sin^{5}(a) - 9120\, \sin^{3}(a) + 38\, \sin(a)\big) \cdot \cos(a)$ $\sin(39a) = - 274877906944\, \sin^{39}(a) + 2680059592704\, \sin^{37}(a) - 12060268167168\, \sin^{35}(a) + 33221572034560\, \sin^{33}(a) - 62646392979456\, \sin^{31}(a) + 85678155104256\, \sin^{29}(a) - 87841744879616\, \sin^{27}(a) + 68822438510592\, \sin^{25}(a) - 41626474905600\, \sin^{23}(a) + 19502774353920\, \sin^{21}(a) - 7061349335040\, \sin^{19}(a) + 1960212234240\, \sin^{17}(a) - 411402567680\, \sin^{15}(a) + 63901286400\, \sin^{13}(a) - 7120429056\, \sin^{11}(a) + 543921664\, \sin^{9}(a) - 26604864\, \sin^{7}(a) + 746928\, \sin^{5}(a) - 9880\, \sin^{3}(a) + 39\, \sin(a)$ $\sin(40a) = \big( - 549755813888\, \sin^{39}(a) + 5222680231936\, \sin^{37}(a) - 22883585753088\, \sin^{35}(a) + 61332132986880\, \sin^{33}(a) - 112442243809280\, \sin^{31}(a) + 149387552489472\, \sin^{29}(a) - 148655260565504\, \sin^{27}(a) + 112939386273792\, \sin^{25}(a) - 66175421644800\, \sin^{23}(a) + 30004268236800\, \sin^{21}(a) - 10501493882880\, \sin^{19}(a) + 2814663720960\, \sin^{17}(a) - 569634324480\, \sin^{15}(a) + 85201715200\, \sin^{13}(a) - 9128755200\, \sin^{11}(a) + 669442048\, \sin^{9}(a) - 31380096\, \sin^{7}(a) + 842688\, \sin^{5}(a) - 10640\, \sin^{3}(a) + 40\, \sin(a)\big) \cdot \cos(a)$ $\sin(41a) = 1099511627776\, \sin^{41}(a) - 11269994184704\, \sin^{39}(a) + 53532472377344\, \sin^{37}(a) - 156371169312768\, \sin^{35}(a) + 314327181557760\, \sin^{33}(a) - 461013199618048\, \sin^{31}(a) + 510407471005696\, \sin^{29}(a) - 435347548798976\, \sin^{27}(a) + 289407177326592\, \sin^{25}(a) - 150732904857600\, \sin^{23}(a) + 61508749885440\, \sin^{21}(a) - 19570965872640\, \sin^{19}(a) + 4808383856640\, \sin^{17}(a) - 898269511680\, \sin^{15}(a) + 124759654400\, \sin^{13}(a) - 12475965440\, \sin^{11}(a) + 857722624\, \sin^{9}(a) - 37840704\, \sin^{7}(a) + 959728\, \sin^{5}(a) - 11480\, \sin^{3}(a) + 41\, \sin(a)$ $\sin(42a) = \big(2199023255552\, \sin^{41}(a) - 21990232555520\, \sin^{39}(a) + 101842264522752\, \sin^{37}(a) - 289858752872448\, \sin^{35}(a) + 567322230128640\, \sin^{33}(a) - 809584155426816\, \sin^{31}(a) + 871427389521920\, \sin^{29}(a) - 722039837032448\, \sin^{27}(a) + 465874968379392\, \sin^{25}(a) - 235290388070400\, \sin^{23}(a) + 93013231534080\, \sin^{21}(a) - 28640437862400\, \sin^{19}(a) + 6802103992320\, \sin^{17}(a) - 1226904698880\, \sin^{15}(a) + 164317593600\, \sin^{13}(a) - 15823175680\, \sin^{11}(a) + 1046003200\, \sin^{9}(a) - 44301312\, \sin^{7}(a) + 1076768\, \sin^{5}(a) - 12320\, \sin^{3}(a) + 42\, \sin(a)\big) \cdot \cos(a)$ $\sin(43a) = - 4398046511104\, \sin^{43}(a) + 47278999994368\, \sin^{41}(a) - 236394999971840\, \sin^{39}(a) + 729869562413056\, \sin^{37}(a) - 1557990796689408\, \sin^{35}(a) + 2439485589553152\, \sin^{33}(a) - 2901009890279424\, \sin^{31}(a) + 2676526982103040\, \sin^{29}(a) - 1940482062024704\, \sin^{27}(a) + 1112923535572992\, \sin^{25}(a) - 505874334351360\, \sin^{23}(a) + 181798588907520\, \sin^{21}(a) - 51314117836800\, \sin^{19}(a) + 11249633525760\, \sin^{17}(a) - 1884175073280\, \sin^{15}(a) + 235521884160\, \sin^{13}(a) - 21262392320\, \sin^{11}(a) + 1322886400\, \sin^{9}(a) - 52915456\, \sin^{7}(a) + 1218448\, \sin^{5}(a) - 13244\, \sin^{3}(a) + 43\, \sin(a)$ $\sin(44a) = \big( - 8796093022208\, \sin^{43}(a) + 92358976733184\, \sin^{41}(a) - 450799767388160\, \sin^{39}(a) + 1357896860303360\, \sin^{37}(a) - 2826122840506368\, \sin^{35}(a) + 4311648948977664\, \sin^{33}(a) - 4992435625132032\, \sin^{31}(a) + 4481626574684160\, \sin^{29}(a) - 3158924287016960\, \sin^{27}(a) + 1759972102766592\, \sin^{25}(a) - 776458280632320\, \sin^{23}(a) + 270583946280960\, \sin^{21}(a) - 73987797811200\, \sin^{19}(a) + 15697163059200\, \sin^{17}(a) - 2541445447680\, \sin^{15}(a) + 306726174720\, \sin^{13}(a) - 26701608960\, \sin^{11}(a) + 1599769600\, \sin^{9}(a) - 61529600\, \sin^{7}(a) + 1360128\, \sin^{5}(a) - 14168\, \sin^{3}(a) + 44\, \sin(a)\big) \cdot \cos(a)$ $\sin(45a) = 17592186044416\, \sin^{45}(a) - 197912092999680\, \sin^{43}(a) + 1039038488248320\, \sin^{41}(a) - 3380998255411200\, \sin^{39}(a) + 7638169839206400\, \sin^{37}(a) - 12717552782278656\, \sin^{35}(a) + 16168683558666240\, \sin^{33}(a) - 16047114509352960\, \sin^{31}(a) + 12604574741299200\, \sin^{29}(a) - 7897310717542400\, \sin^{27}(a) + 3959937231224832\, \sin^{25}(a) - 1588210119475200\, \sin^{23}(a) + 507344899276800\, \sin^{21}(a) - 128055803904000\, \sin^{19}(a) + 25227583488000\, \sin^{17}(a) - 3812168171520\, \sin^{15}(a) + 431333683200\, \sin^{13}(a) - 35340364800\, \sin^{11}(a) + 1999712000\, \sin^{9}(a) - 72864000\, \sin^{7}(a) + 1530144\, \sin^{5}(a) - 15180\, \sin^{3}(a) + 45\, \sin(a)$ $\sin(46a) = \big(35184372088832\, \sin^{45}(a) - 387028092977152\, \sin^{43}(a) + 1985717999763456\, \sin^{41}(a) - 6311196743434240\, \sin^{39}(a) + 13918442818109440\, \sin^{37}(a) - 22608982724050944\, \sin^{35}(a) + 28025718168354816\, \sin^{33}(a) - 27101793393573888\, \sin^{31}(a) + 20727522907914240\, \sin^{29}(a) - 12635697148067840\, \sin^{27}(a) + 6159902359683072\, \sin^{25}(a) - 2399961958318080\, \sin^{23}(a) + 744105852272640\, \sin^{21}(a) - 182123809996800\, \sin^{19}(a) + 34758003916800\, \sin^{17}(a) - 5082890895360\, \sin^{15}(a) + 555941191680\, \sin^{13}(a) - 43979120640\, \sin^{11}(a) + 2399654400\, \sin^{9}(a) - 84198400\, \sin^{7}(a) + 1700160\, \sin^{5}(a) - 16192\, \sin^{3}(a) + 46\, \sin(a)\big) \cdot \cos(a)$ $\sin(47a) = - 70368744177664\, \sin^{47}(a) + 826832744087552\, \sin^{45}(a) - 4547580092481536\, \sin^{43}(a) + 15554790998147072\, \sin^{41}(a) - 37078280867676160\, \sin^{39}(a) + 65416681245114368\, \sin^{37}(a) - 88551849002532864\, \sin^{35}(a) + 94086339565191168\, \sin^{33}(a) - 79611518093623296\, \sin^{31}(a) + 54121865370664960\, \sin^{29}(a) - 29693888297959424\, \sin^{27}(a) + 13159791404777472\, \sin^{25}(a) - 4699925501706240\, \sin^{23}(a) + 1345114425262080\, \sin^{21}(a) - 305707823923200\, \sin^{19}(a) + 54454206136320\, \sin^{17}(a) - 7465496002560\, \sin^{15}(a) + 768506941440\, \sin^{13}(a) - 57417185280\, \sin^{11}(a) + 2967993600\, \sin^{9}(a) - 98933120\, \sin^{7}(a) + 1902560\, \sin^{5}(a) - 17296\, \sin^{3}(a) + 47\, \sin(a)$ $\sin(48a) = \big( - 140737488355328\, \sin^{47}(a) + 1618481116086272\, \sin^{45}(a) - 8708132091985920\, \sin^{43}(a) + 29123863996530688\, \sin^{41}(a) - 67845364991918080\, \sin^{39}(a) + 116914919672119296\, \sin^{37}(a) - 154494715281014784\, \sin^{35}(a) + 160146960962027520\, \sin^{33}(a) - 132121242793672704\, \sin^{31}(a) + 87516207833415680\, \sin^{29}(a) - 46752079447851008\, \sin^{27}(a) + 20159680449871872\, \sin^{25}(a) - 6999889045094400\, \sin^{23}(a) + 1946122998251520\, \sin^{21}(a) - 429291837849600\, \sin^{19}(a) + 74150408355840\, \sin^{17}(a) - 9848101109760\, \sin^{15}(a) + 981072691200\, \sin^{13}(a) - 70855249920\, \sin^{11}(a) + 3536332800\, \sin^{9}(a) - 113667840\, \sin^{7}(a) + 2104960\, \sin^{5}(a) - 18400\, \sin^{3}(a) + 48\, \sin(a)\big) \cdot \cos(a)$ $\sin(49a) = 281474976710656\, \sin^{49}(a) - 3448068464705536\, \sin^{47}(a) + 19826393672056832\, \sin^{45}(a) - 71116412084551680\, \sin^{43}(a) + 178383666978750464\, \sin^{41}(a) - 332442288460398592\, \sin^{39}(a) + 477402588661153792\, \sin^{37}(a) - 540731503483551744\, \sin^{35}(a) + 490450067946209280\, \sin^{33}(a) - 359663383160553472\, \sin^{31}(a) + 214414709191868416\, \sin^{29}(a) - 104129631497486336\, \sin^{27}(a) + 41159347585155072\, \sin^{25}(a) - 13192098584985600\, \sin^{23}(a) + 3405715246940160\, \sin^{21}(a) - 701176668487680\, \sin^{19}(a) + 113542812794880\, \sin^{17}(a) - 14192851599360\, \sin^{15}(a) + 1335348940800\, \sin^{13}(a) - 91365980160\, \sin^{11}(a) + 4332007680\, \sin^{9}(a) - 132612480\, \sin^{7}(a) + 2344160\, \sin^{5}(a) - 19600\, \sin^{3}(a) + 49\, \sin(a)$ $\sin(50a) = \big(562949953421312\, \sin^{49}(a) - 6755399441055744\, \sin^{47}(a) + 38034306228027392\, \sin^{45}(a) - 133524692077117440\, \sin^{43}(a) + 327643469960970240\, \sin^{41}(a) - 597039211928879104\, \sin^{39}(a) + 837890257650188288\, \sin^{37}(a) - 926968291686088704\, \sin^{35}(a) + 820753174930391040\, \sin^{33}(a) - 587205523527434240\, \sin^{31}(a) + 341313210550321152\, \sin^{29}(a) - 161507183547121664\, \sin^{27}(a) + 62159014720438272\, \sin^{25}(a) - 19384308124876800\, \sin^{23}(a) + 4865307495628800\, \sin^{21}(a) - 973061499125760\, \sin^{19}(a) + 152935217233920\, \sin^{17}(a) - 18537602088960\, \sin^{15}(a) + 1689625190400\, \sin^{13}(a) - 111876710400\, \sin^{11}(a) + 5127682560\, \sin^{9}(a) - 151557120\, \sin^{7}(a) + 2583360\, \sin^{5}(a) - 20800\, \sin^{3}(a) + 50\, \sin(a)\big) \cdot \cos(a)$ $\sin(51a) = - 1125899906842624\, \sin^{51}(a) + 14355223812243456\, \sin^{49}(a) - 86131342873460736\, \sin^{47}(a) + 323291602938232832\, \sin^{45}(a) - 851219911991623680\, \sin^{43}(a) + 1670981696800948224\, \sin^{41}(a) - 2537416650697736192\, \sin^{39}(a) + 3052314510011400192\, \sin^{37}(a) - 2954711429749407744\, \sin^{35}(a) + 2325467328969441280\, \sin^{33}(a) - 1497374084994957312\, \sin^{31}(a) + 791226079003017216\, \sin^{29}(a) - 343202765037633536\, \sin^{27}(a) + 121927298105475072\, \sin^{25}(a) - 35307132656025600\, \sin^{23}(a) + 8271022742568960\, \sin^{21}(a) - 1550816764231680\, \sin^{19}(a) + 229402825850880\, \sin^{17}(a) - 26261602959360\, \sin^{15}(a) + 2267654860800\, \sin^{13}(a) - 142642805760\, \sin^{11}(a) + 6226471680\, \sin^{9}(a) - 175668480\, \sin^{7}(a) + 2864160\, \sin^{5}(a) - 22100\, \sin^{3}(a) + 51\, \sin(a)$ $\sin(52a) = \big( - 2251799813685248\, \sin^{51}(a) + 28147497671065600\, \sin^{49}(a) - 165507286305865728\, \sin^{47}(a) + 608548899648438272\, \sin^{45}(a) - 1568915131906129920\, \sin^{43}(a) + 3014319923640926208\, \sin^{41}(a) - 4477794089466593280\, \sin^{39}(a) + 5266738762372612096\, \sin^{37}(a) - 4982454567812726784\, \sin^{35}(a) + 3830181483008491520\, \sin^{33}(a) - 2407542646462480384\, \sin^{31}(a) + 1241138947455713280\, \sin^{29}(a) - 524898346528145408\, \sin^{27}(a) + 181695581490511872\, \sin^{25}(a) - 51229957187174400\, \sin^{23}(a) + 11676737989509120\, \sin^{21}(a) - 2128572029337600\, \sin^{19}(a) + 305870434467840\, \sin^{17}(a) - 33985603829760\, \sin^{15}(a) + 2845684531200\, \sin^{13}(a) - 173408901120\, \sin^{11}(a) + 7325260800\, \sin^{9}(a) - 199779840\, \sin^{7}(a) + 3144960\, \sin^{5}(a) - 23400\, \sin^{3}(a) + 52\, \sin(a)\big) \cdot \cos(a)$ $\sin(53a) = 4503599627370496\, \sin^{53}(a) - 59672695062659072\, \sin^{51}(a) + 372954344141619200\, \sin^{49}(a) - 1461981029035147264\, \sin^{47}(a) + 4031636460170903552\, \sin^{45}(a) - 8315250199102488576\, \sin^{43}(a) + 13313246329414090752\, \sin^{41}(a) - 16951649052980674560\, \sin^{39}(a) + 17446072150359277568\, \sin^{37}(a) - 14670560671893028864\, \sin^{35}(a) + 10149980929972502528\, \sin^{33}(a) - 5799989102841430016\, \sin^{31}(a) + 2740848508964700160\, \sin^{29}(a) - 1069985090999681024\, \sin^{27}(a) + 343923779249897472\, \sin^{25}(a) - 90506257697341440\, \sin^{23}(a) + 19339597295124480\, \sin^{21}(a) - 3318068163379200\, \sin^{19}(a) + 450309250744320\, \sin^{17}(a) - 47400973762560\, \sin^{15}(a) + 3770532003840\, \sin^{13}(a) - 218825518080\, \sin^{11}(a) + 8823609600\, \sin^{9}(a) - 230181120\, \sin^{7}(a) + 3472560\, \sin^{5}(a) - 24804\, \sin^{3}(a) + 53\, \sin(a)$ $\sin(54a) = \big(9007199254740992\, \sin^{53}(a) - 117093590311632896\, \sin^{51}(a) + 717761190612172800\, \sin^{49}(a) - 2758454771764428800\, \sin^{47}(a) + 7454724020693368832\, \sin^{45}(a) - 15061585266298847232\, \sin^{43}(a) + 23612172735187255296\, \sin^{41}(a) - 29425504016494755840\, \sin^{39}(a) + 29625405538345943040\, \sin^{37}(a) - 24358666775973330944\, \sin^{35}(a) + 16469780376936513536\, \sin^{33}(a) - 9192435559220379648\, \sin^{31}(a) + 4240558070473687040\, \sin^{29}(a) - 1615071835471216640\, \sin^{27}(a) + 506151977009283072\, \sin^{25}(a) - 129782558207508480\, \sin^{23}(a) + 27002456600739840\, \sin^{21}(a) - 4507564297420800\, \sin^{19}(a) + 594748067020800\, \sin^{17}(a) - 60816343695360\, \sin^{15}(a) + 4695379476480\, \sin^{13}(a) - 264242135040\, \sin^{11}(a) + 10321958400\, \sin^{9}(a) - 260582400\, \sin^{7}(a) + 3800160\, \sin^{5}(a) - 26208\, \sin^{3}(a) + 54\, \sin(a)\big) \cdot \cos(a)$ $\sin(55a) = - 18014398509481984\, \sin^{55}(a) + 247697979505377280\, \sin^{53}(a) - 1610036866784952320\, \sin^{51}(a) + 6579477580611584000\, \sin^{49}(a) - 18964376555880448000\, \sin^{47}(a) + 41000982113813528576\, \sin^{45}(a) - 69032265803869716480\, \sin^{43}(a) + 92762107173949931520\, \sin^{41}(a) - 101150170056700723200\, \sin^{39}(a) + 90522072478279270400\, \sin^{37}(a) - 66986333633926660096\, \sin^{35}(a) + 41174450942341283840\, \sin^{33}(a) - 21065998156546703360\, \sin^{31}(a) + 8970411302925107200\, \sin^{29}(a) - 3172462533961318400\, \sin^{27}(a) + 927945291183685632\, \sin^{25}(a) - 223063771919155200\, \sin^{23}(a) + 43680444501196800\, \sin^{21}(a) - 6886556565504000\, \sin^{19}(a) + 860819570688000\, \sin^{17}(a) - 83622472581120\, \sin^{15}(a) + 6148711219200\, \sin^{13}(a) - 330302668800\, \sin^{11}(a) + 12341472000\, \sin^{9}(a) - 298584000\, \sin^{7}(a) + 4180176\, \sin^{5}(a) - 27720\, \sin^{3}(a) + 55\, \sin(a)$ $\sin(56a) = \big( - 36028797018963968\, \sin^{55}(a) + 486388759756013568\, \sin^{53}(a) - 3102980143258271744\, \sin^{51}(a) + 12441193970610995200\, \sin^{49}(a) - 35170298339996467200\, \sin^{47}(a) + 74547240206933688320\, \sin^{45}(a) - 123002946341440585728\, \sin^{43}(a) + 161912041612712607744\, \sin^{41}(a) - 172874836096906690560\, \sin^{39}(a) + 151418739418212597760\, \sin^{37}(a) - 109614000491879989248\, \sin^{35}(a) + 65879121507746054144\, \sin^{33}(a) - 32939560753873027072\, \sin^{31}(a) + 13700264535376527360\, \sin^{29}(a) - 4729853232451420160\, \sin^{27}(a) + 1349738605358088192\, \sin^{25}(a) - 316344985630801920\, \sin^{23}(a) + 60358432401653760\, \sin^{21}(a) - 9265548833587200\, \sin^{19}(a) + 1126891074355200\, \sin^{17}(a) - 106428601466880\, \sin^{15}(a) + 7602042961920\, \sin^{13}(a) - 396363202560\, \sin^{11}(a) + 14360985600\, \sin^{9}(a) - 336585600\, \sin^{7}(a) + 4560192\, \sin^{5}(a) - 29232\, \sin^{3}(a) + 56\, \sin(a)\big) \cdot \cos(a)$ $\sin(57a) = 72057594037927936\, \sin^{57}(a) - 1026820715040473088\, \sin^{55}(a) + 6931039826523193344\, \sin^{53}(a) - 29478311360953581568\, \sin^{51}(a) + 88643507040603340800\, \sin^{49}(a) - 200470700537979863040\, \sin^{47}(a) + 354099390982935019520\, \sin^{45}(a) - 500797710104436670464\, \sin^{43}(a) + 576811648245288665088\, \sin^{41}(a) - 547436980973537853440\, \sin^{39}(a) + 431543407341905903616\, \sin^{37}(a) - 283999910365325426688\, \sin^{35}(a) + 156462913580896878592\, \sin^{33}(a) - 72213652421952405504\, \sin^{31}(a) + 27889824232730787840\, \sin^{29}(a) - 8986721141657698304\, \sin^{27}(a) + 2404221890794094592\, \sin^{25}(a) - 530343064145756160\, \sin^{23}(a) + 95567517969285120\, \sin^{21}(a) - 13898323250380800\, \sin^{19}(a) + 1605819780956160\, \sin^{17}(a) - 144438816276480\, \sin^{15}(a) + 9848101109760\, \sin^{13}(a) - 491145707520\, \sin^{11}(a) + 17053670400\, \sin^{9}(a) - 383707584\, \sin^{7}(a) + 4998672\, \sin^{5}(a) - 30856\, \sin^{3}(a) + 57\, \sin(a)$ $\sin(58a) = \big(144115188075855872\, \sin^{57}(a) - 2017612633061982208\, \sin^{55}(a) + 13375690893290373120\, \sin^{53}(a) - 55853642578648891392\, \sin^{51}(a) + 164845820110595686400\, \sin^{49}(a) - 365771102735963258880\, \sin^{47}(a) + 633651541758936350720\, \sin^{45}(a) - 878592473867432755200\, \sin^{43}(a) + 991711254877864722432\, \sin^{41}(a) - 921999125850169016320\, \sin^{39}(a) + 711668075265599209472\, \sin^{37}(a) - 458385820238770864128\, \sin^{35}(a) + 247046705654047703040\, \sin^{33}(a) - 111487744090031783936\, \sin^{31}(a) + 42079383930085048320\, \sin^{29}(a) - 13243589050863976448\, \sin^{27}(a) + 3458705176230100992\, \sin^{25}(a) - 744341142660710400\, \sin^{23}(a) + 130776603536916480\, \sin^{21}(a) - 18531097667174400\, \sin^{19}(a) + 2084748487557120\, \sin^{17}(a) - 182449031086080\, \sin^{15}(a) + 12094159257600\, \sin^{13}(a) - 585928212480\, \sin^{11}(a) + 19746355200\, \sin^{9}(a) - 430829568\, \sin^{7}(a) + 5437152\, \sin^{5}(a) - 32480\, \sin^{3}(a) + 58\, \sin(a)\big) \cdot \cos(a)$ $\sin(59a) = - 288230376151711744\, \sin^{59}(a) + 4251398048237748224\, \sin^{57}(a) - 29759786337664237568\, \sin^{55}(a) + 131527627117355335680\, \sin^{53}(a) - 411920614017535574016\, \sin^{51}(a) + 972590338652514549760\, \sin^{49}(a) - 1798374588451819356160\, \sin^{47}(a) + 2670388640269803192320\, \sin^{45}(a) - 3239809747386158284800\, \sin^{43}(a) + 3250609113210778812416\, \sin^{41}(a) - 2719897421257998598144\, \sin^{39}(a) + 1908564383666834243584\, \sin^{37}(a) - 1126865141420311707648\, \sin^{35}(a) + 560605985907262095360\, \sin^{33}(a) - 234920603618281259008\, \sin^{31}(a) + 82756121729167261696\, \sin^{29}(a) - 24417867312530456576\, \sin^{27}(a) + 6001870746987528192\, \sin^{25}(a) - 1219892428249497600\, \sin^{23}(a) + 203047884438896640\, \sin^{21}(a) - 27333369059082240\, \sin^{19}(a) + 2928575256330240\, \sin^{17}(a) - 244647564410880\, \sin^{15}(a) + 15512073830400\, \sin^{13}(a) - 720203427840\, \sin^{11}(a) + 23300699136\, \sin^{9}(a) - 488825856\, \sin^{7}(a) + 5940592\, \sin^{5}(a) - 34220\, \sin^{3}(a) + 59\, \sin(a)$ $\sin(60a) = \big( - 576460752303423488\, \sin^{59}(a) + 8358680908399640576\, \sin^{57}(a) - 57501960042266492928\, \sin^{55}(a) + 249679563341420298240\, \sin^{53}(a) - 767987585456422256640\, \sin^{51}(a) + 1780334857194433413120\, \sin^{49}(a) - 3230978074167675453440\, \sin^{47}(a) + 4707125738780670033920\, \sin^{45}(a) - 5601027020904883814400\, \sin^{43}(a) + 5509506971543692902400\, \sin^{41}(a) - 4517795716665828179968\, \sin^{39}(a) + 3105460692068069277696\, \sin^{37}(a) - 1795344462601852551168\, \sin^{35}(a) + 874165266160476487680\, \sin^{33}(a) - 358353463146530734080\, \sin^{31}(a) + 123432859528249475072\, \sin^{29}(a) - 35592145574196936704\, \sin^{27}(a) + 8545036317744955392\, \sin^{25}(a) - 1695443713838284800\, \sin^{23}(a) + 275319165340876800\, \sin^{21}(a) - 36135640450990080\, \sin^{19}(a) + 3772402025103360\, \sin^{17}(a) - 306846097735680\, \sin^{15}(a) + 18929988403200\, \sin^{13}(a) - 854478643200\, \sin^{11}(a) + 26855043072\, \sin^{9}(a) - 546822144\, \sin^{7}(a) + 6444032\, \sin^{5}(a) - 35960\, \sin^{3}(a) + 60\, \sin(a)\big) \cdot \cos(a)$ $\sin(61a) = 1152921504606846976\, \sin^{61}(a) - 17582052945254416384\, \sin^{59}(a) + 127469883853094518784\, \sin^{57}(a) - 584603260429709344768\, \sin^{55}(a) + 1903806670478329774080\, \sin^{53}(a) - 4684724271284175765504\, \sin^{51}(a) + 9050035524071703183360\, \sin^{49}(a) - 14077833037444871618560\, \sin^{47}(a) + 17945916879101304504320\, \sin^{45}(a) - 18981258237510995148800\, \sin^{43}(a) + 16803996263208263352320\, \sin^{41}(a) - 12526615396209796317184\, \sin^{39}(a) + 7893045925673009414144\, \sin^{37}(a) - 4212154316104346370048\, \sin^{35}(a) + 1904431472706752348160\, \sin^{33}(a) - 728652041731279159296\, \sin^{31}(a) + 235293888475725561856\, \sin^{29}(a) - 63856496471353327616\, \sin^{27}(a) + 14479089316178952192\, \sin^{25}(a) - 2721633330108825600\, \sin^{23}(a) + 419861727144837120\, \sin^{21}(a) - 52482715893104640\, \sin^{19}(a) + 5229920989347840\, \sin^{17}(a) - 406904607866880\, \sin^{15}(a) + 24056860262400\, \sin^{13}(a) - 1042463944704\, \sin^{11}(a) + 31503031296\, \sin^{9}(a) - 617706496\, \sin^{7}(a) + 7019392\, \sin^{5}(a) - 37820\, \sin^{3}(a) + 61\, \sin(a)$ $\sin(62a) = \big(2305843009213693952\, \sin^{61}(a) - 34587645138205409280\, \sin^{59}(a) + 246581086797789396992\, \sin^{57}(a) - 1111704560817152196608\, \sin^{55}(a) + 3557933777615239249920\, \sin^{53}(a) - 8601460957111929274368\, \sin^{51}(a) + 16319736190948972953600\, \sin^{49}(a) - 24924688000722067783680\, \sin^{47}(a) + 31184708019421938974720\, \sin^{45}(a) - 32361489454117106483200\, \sin^{43}(a) + 28098485554872833802240\, \sin^{41}(a) - 20535435075753764454400\, \sin^{39}(a) + 12680631159277949550592\, \sin^{37}(a) - 6628964169606840188928\, \sin^{35}(a) + 2934697679253028208640\, \sin^{33}(a) - 1098950620316027584512\, \sin^{31}(a) + 347154917423201648640\, \sin^{29}(a) - 92120847368509718528\, \sin^{27}(a) + 20413142314612948992\, \sin^{25}(a) - 3747822946379366400\, \sin^{23}(a) + 564404288948797440\, \sin^{21}(a) - 68829791335219200\, \sin^{19}(a) + 6687439953592320\, \sin^{17}(a) - 506963117998080\, \sin^{15}(a) + 29183732121600\, \sin^{13}(a) - 1230449246208\, \sin^{11}(a) + 36151019520\, \sin^{9}(a) - 688590848\, \sin^{7}(a) + 7594752\, \sin^{5}(a) - 39680\, \sin^{3}(a) + 62\, \sin(a)\big) \cdot \cos(a)$ $\sin(63a) = - 4611686018427387904\, \sin^{63}(a) + 72634054790231359488\, \sin^{61}(a) - 544755410926735196160\, \sin^{59}(a) + 2589101411376788668416\, \sin^{57}(a) - 8754673416435073548288\, \sin^{55}(a) + 22414982798976007274496\, \sin^{53}(a) - 45157670024837628690432\, \sin^{51}(a) + 73438812859270378291200\, \sin^{49}(a) - 98140959002843141898240\, \sin^{47}(a) + 109146478067976786411520\, \sin^{45}(a) - 101938691780468885422080\, \sin^{43}(a) + 80463844998044933160960\, \sin^{41}(a) - 53905517073853631692800\, \sin^{39}(a) + 30726144732096570064896\, \sin^{37}(a) - 14915169381615390425088\, \sin^{35}(a) + 6162865126431359238144\, \sin^{33}(a) - 2163559033747179307008\, \sin^{31}(a) + 643257641107697172480\, \sin^{29}(a) - 161211482894892007424\, \sin^{27}(a) + 33842841205805678592\, \sin^{25}(a) - 5902821140547502080\, \sin^{23}(a) + 846606433423196160\, \sin^{21}(a) - 98551746684518400\, \sin^{19}(a) + 9158885153832960\, \sin^{17}(a) - 665389092372480\, \sin^{15}(a) + 36771502473216\, \sin^{13}(a) - 1490736586752\, \sin^{11}(a) + 42176189440\, \sin^{9}(a) - 774664704\, \sin^{7}(a) + 8249472\, \sin^{5}(a) - 41664\, \sin^{3}(a) + 63\, \sin(a)$ $\sin(64a) = \big( - 9223372036854775808\, \sin^{63}(a) + 142962266571249025024\, \sin^{61}(a) - 1054923176715264983040\, \sin^{59}(a) + 4931621735955787939840\, \sin^{57}(a) - 16397642272052994899968\, \sin^{55}(a) + 41272031820336775299072\, \sin^{53}(a) - 81713879092563328106496\, \sin^{51}(a) + 130557889527591783628800\, \sin^{49}(a) - 171357230004964216012800\, \sin^{47}(a) + 187108248116531633848320\, \sin^{45}(a) - 171515894106820664360960\, \sin^{43}(a) + 132829204441217032519680\, \sin^{41}(a) - 87275599071953498931200\, \sin^{39}(a) + 48771658304915190579200\, \sin^{37}(a) - 23201374593623940661248\, \sin^{35}(a) + 9391032573609690267648\, \sin^{33}(a) - 3228167447178331029504\, \sin^{31}(a) + 939360364792192696320\, \sin^{29}(a) - 230302118421274296320\, \sin^{27}(a) + 47272540096998408192\, \sin^{25}(a) - 8057819334715637760\, \sin^{23}(a) + 1128808577897594880\, \sin^{21}(a) - 128273702033817600\, \sin^{19}(a) + 11630330354073600\, \sin^{17}(a) - 823815066746880\, \sin^{15}(a) + 44359272824832\, \sin^{13}(a) - 1751023927296\, \sin^{11}(a) + 48201359360\, \sin^{9}(a) - 860738560\, \sin^{7}(a) + 8904192\, \sin^{5}(a) - 43648\, \sin^{3}(a) + 64\, \sin(a)\big) \cdot \cos(a)$ $\sin(65a) = 18446744073709551616\, \sin^{65}(a) - 299759591197780213760\, \sin^{63}(a) + 2323136831782796656640\, \sin^{61}(a) - 11428334414415370649600\, \sin^{59}(a) + 40069426604640777011200\, \sin^{57}(a) - 106584674768344466849792\, \sin^{55}(a) + 223556839026824199536640\, \sin^{53}(a) - 379385867215472594780160\, \sin^{51}(a) + 530391426205841620992000\, \sin^{49}(a) - 618789997240148557824000\, \sin^{47}(a) + 608101806378727810007040\, \sin^{45}(a) - 506751505315606508339200\, \sin^{43}(a) + 359745762028296129740800\, \sin^{41}(a) - 218188997679883747328000\, \sin^{39}(a) + 113219921064981692416000\, \sin^{37}(a) - 50269644952851871432704\, \sin^{35}(a) + 19075534915144683356160\, \sin^{33}(a) - 6171496590193868144640\, \sin^{31}(a) + 1696067325319236812800\, \sin^{29}(a) - 393937834141653401600\, \sin^{27}(a) + 76817877657622413312\, \sin^{25}(a) - 12470434684678963200\, \sin^{23}(a) + 1667558126439628800\, \sin^{21}(a) - 181256318091264000\, \sin^{19}(a) + 15749405687808000\, \sin^{17}(a) - 1070959586770944\, \sin^{15}(a) + 55449091031040\, \sin^{13}(a) - 2107713986560\, \sin^{11}(a) + 55948006400\, \sin^{9}(a) - 964620800\, \sin^{7}(a) + 9646208\, \sin^{5}(a) - 45760\, \sin^{3}(a) + 65\, \sin(a)$ $\sin(66a) = \big(36893488147419103232\, \sin^{65}(a) - 590295810358705651712\, \sin^{63}(a) + 4503311396994344288256\, \sin^{61}(a) - 21801745652115476316160\, \sin^{59}(a) + 75207231473325766082560\, \sin^{57}(a) - 196771707264635938799616\, \sin^{55}(a) + 405841646233311623774208\, \sin^{53}(a) - 677057855338381861453824\, \sin^{51}(a) + 930224962884091458355200\, \sin^{49}(a) - 1066222764475332899635200\, \sin^{47}(a) + 1029095364640923986165760\, \sin^{45}(a) - 841987116524392352317440\, \sin^{43}(a) + 586662319615375226961920\, \sin^{41}(a) - 349102396287813995724800\, \sin^{39}(a) + 177668183825048194252800\, \sin^{37}(a) - 77337915312079802204160\, \sin^{35}(a) + 28760037256679676444672\, \sin^{33}(a) - 9114825733209405259776\, \sin^{31}(a) + 2452774285846280929280\, \sin^{29}(a) - 557573549862032506880\, \sin^{27}(a) + 106363215218246418432\, \sin^{25}(a) - 16883050034642288640\, \sin^{23}(a) + 2206307674981662720\, \sin^{21}(a) - 234238934148710400\, \sin^{19}(a) + 19868481021542400\, \sin^{17}(a) - 1318104106795008\, \sin^{15}(a) + 66538909237248\, \sin^{13}(a) - 2464404045824\, \sin^{11}(a) + 63694653440\, \sin^{9}(a) - 1068503040\, \sin^{7}(a) + 10388224\, \sin^{5}(a) - 47872\, \sin^{3}(a) + 66\, \sin(a)\big) \cdot \cos(a)$ $\sin(67a) = - 73786976294838206464\, \sin^{67}(a) + 1235931852938539958272\, \sin^{65}(a) - 9887454823508319666176\, \sin^{63}(a) + 50286977266436844552192\, \sin^{61}(a) - 182589619836467114147840\, \sin^{59}(a) + 503888450871282632753152\, \sin^{57}(a) - 1098642032227550658297856\, \sin^{55}(a) + 1942242164116562770919424\, \sin^{53}(a) - 2835179769229474044837888\, \sin^{51}(a) + 3462504028513007094988800\, \sin^{49}(a) - 3571846260992365213777920\, \sin^{47}(a) + 3134063155951904866959360\, \sin^{45}(a) - 2350547366963928650219520\, \sin^{43}(a) + 1511783669778082315632640\, \sin^{41}(a) - 835352162545840632627200\, \sin^{39}(a) + 396792277209274300497920\, \sin^{37}(a) - 161926260184667085864960\, \sin^{35}(a) + 56674191064633480052736\, \sin^{33}(a) - 16963703447917504233472\, \sin^{31}(a) + 4324628346097390059520\, \sin^{29}(a) - 933935696018904449024\, \sin^{27}(a) + 169674652848155000832\, \sin^{25}(a) - 25708280734568939520\, \sin^{23}(a) + 3213535091821117440\, \sin^{21}(a) - 326958512249241600\, \sin^{19}(a) + 26623764568866816\, \sin^{17}(a) - 1698326445293568\, \sin^{15}(a) + 82557535535104\, \sin^{13}(a) - 2948483411968\, \sin^{11}(a) + 73578306560\, \sin^{9}(a) - 1193161728\, \sin^{7}(a) + 11225984\, \sin^{5}(a) - 50116\, \sin^{3}(a) + 67\, \sin(a)$ $\sin(68a) = \big( - 147573952589676412928\, \sin^{67}(a) + 2434970217729660813312\, \sin^{65}(a) - 19184613836657933680640\, \sin^{63}(a) + 96070643135879344816128\, \sin^{61}(a) - 343377494020818751979520\, \sin^{59}(a) + 932569670269239499423744\, \sin^{57}(a) - 2000512357190465377796096\, \sin^{55}(a) + 3478642681999813918064640\, \sin^{53}(a) - 4993301683120566228221952\, \sin^{51}(a) + 5994783094141922731622400\, \sin^{49}(a) - 6077469757509397527920640\, \sin^{47}(a) + 5239030947262885747752960\, \sin^{45}(a) - 3859107617403464948121600\, \sin^{43}(a) + 2436905019940789404303360\, \sin^{41}(a) - 1321601928803867269529600\, \sin^{39}(a) + 615916370593500406743040\, \sin^{37}(a) - 246514605057254369525760\, \sin^{35}(a) + 84588344872587283660800\, \sin^{33}(a) - 24812581162625603207168\, \sin^{31}(a) + 6196482406348499189760\, \sin^{29}(a) - 1310297842175776391168\, \sin^{27}(a) + 232986090478063583232\, \sin^{25}(a) - 34533511434495590400\, \sin^{23}(a) + 4220762508660572160\, \sin^{21}(a) - 419678090349772800\, \sin^{19}(a) + 33379048116191232\, \sin^{17}(a) - 2078548783792128\, \sin^{15}(a) + 98576161832960\, \sin^{13}(a) - 3432562778112\, \sin^{11}(a) + 83461959680\, \sin^{9}(a) - 1317820416\, \sin^{7}(a) + 12063744\, \sin^{5}(a) - 52360\, \sin^{3}(a) + 68\, \sin(a)\big) \cdot \cos(a)$ $\sin(69a) = 295147905179352825856\, \sin^{69}(a) - 5091301364343836246016\, \sin^{67}(a) + 42003236255836649029632\, \sin^{65}(a) - 220623059121566237327360\, \sin^{63}(a) + 828609297046959349039104\, \sin^{61}(a) - 2369304708743649388658688\, \sin^{59}(a) + 5362275604048127121686528\, \sin^{57}(a) - 9859668046153007933423616\, \sin^{55}(a) + 15001646566124197521653760\, \sin^{53}(a) - 19140989785295503874850816\, \sin^{51}(a) + 20682001674789633424097280\, \sin^{49}(a) - 19061155148552201337569280\, \sin^{47}(a) + 15062213973380796524789760\, \sin^{45}(a) - 10241477907724580054630400\, \sin^{43}(a) + 6005230227711231032033280\, \sin^{41}(a) - 3039684436248894719918080\, \sin^{39}(a) + 1328069674092235252039680\, \sin^{37}(a) - 500279639675016220508160\, \sin^{35}(a) + 162127661005792293683200\, \sin^{33}(a) - 45054423690030700560384\, \sin^{31}(a) + 10688932150951161102336\, \sin^{29}(a) - 2152632169288775499776\, \sin^{27}(a) + 365364550976963346432\, \sin^{25}(a) - 51800267151743385600\, \sin^{23}(a) + 6067346106199572480\, \sin^{21}(a) - 579155764682686464\, \sin^{19}(a) + 44291429231099904\, \sin^{17}(a) - 2655923445956608\, \sin^{15}(a) + 121459913687040\, \sin^{13}(a) - 4083566063616\, \sin^{11}(a) + 95981253632\, \sin^{9}(a) - 1466606592\, \sin^{7}(a) + 13006224\, \sin^{5}(a) - 54740\, \sin^{3}(a) + 69\, \sin(a)$ $\sin(70a) = \big(590295810358705651712\, \sin^{69}(a) - 10035028776097996079104\, \sin^{67}(a) + 81571502293943637245952\, \sin^{65}(a) - 422061504406474540974080\, \sin^{63}(a) + 1561147950958039353262080\, \sin^{61}(a) - 4395231923466480025337856\, \sin^{59}(a) + 9791981537827014743949312\, \sin^{57}(a) - 17718823735115550489051136\, \sin^{55}(a) + 26524650450248581125242880\, \sin^{53}(a) - 33288677887470441521479680\, \sin^{51}(a) + 35369220255437344116572160\, \sin^{49}(a) - 32044840539595005147217920\, \sin^{47}(a) + 24885396999498707301826560\, \sin^{45}(a) - 16623848198045695161139200\, \sin^{43}(a) + 9573555435481672659763200\, \sin^{41}(a) - 4757766943693922170306560\, \sin^{39}(a) + 2040222977590970097336320\, \sin^{37}(a) - 754044674292778071490560\, \sin^{35}(a) + 239666977138997303705600\, \sin^{33}(a) - 65296266217435797913600\, \sin^{31}(a) + 15181381895553823014912\, \sin^{29}(a) - 2994966496401774608384\, \sin^{27}(a) + 497743011475863109632\, \sin^{25}(a) - 69067022868991180800\, \sin^{23}(a) + 7913929703738572800\, \sin^{21}(a) - 738633439015600128\, \sin^{19}(a) + 55203810346008576\, \sin^{17}(a) - 3233298108121088\, \sin^{15}(a) + 144343665541120\, \sin^{13}(a) - 4734569349120\, \sin^{11}(a) + 108500547584\, \sin^{9}(a) - 1615392768\, \sin^{7}(a) + 13948704\, \sin^{5}(a) - 57120\, \sin^{3}(a) + 70\, \sin(a)\big) \cdot \cos(a)$ $\sin(71a) = - 1180591620717411303424\, \sin^{71}(a) + 20955501267734050635776\, \sin^{69}(a) - 178121760775739430404096\, \sin^{67}(a) + 965262777144999707410432\, \sin^{65}(a) - 3745795851607461551144960\, \sin^{63}(a) + 11084150451802079408160768\, \sin^{61}(a) - 26005122213843340149915648\, \sin^{59}(a) + 49659334941837003344314368\, \sin^{57}(a) - 78627280324575255295164416\, \sin^{55}(a) + 104625010109313847771791360\, \sin^{53}(a) - 118174806500520067401252864\, \sin^{51}(a) + 114146119915275065103482880\, \sin^{49}(a) - 94799319929635223560519680\, \sin^{47}(a) + 67956276421708008401141760\, \sin^{45}(a) - 42153329359330155587174400\, \sin^{43}(a) + 22657414530639958628106240\, \sin^{41}(a) - 10556295406320889815367680\, \sin^{39}(a) + 4260465629675261085614080\, \sin^{37}(a) - 1487143663188534529884160\, \sin^{35}(a) + 447798825707073909555200\, \sin^{33}(a) - 115900872535948541296640\, \sin^{31}(a) + 25663764632960034144256\, \sin^{29}(a) - 4832786846466499936256\, \sin^{27}(a) + 768255517712745234432\, \sin^{25}(a) - 102161637993716121600\, \sin^{23}(a) + 11237780179308773376\, \sin^{21}(a) - 1008518734040530944\, \sin^{19}(a) + 72582787677159424\, \sin^{17}(a) - 4099360101367808\, \sin^{15}(a) + 176696556093440\, \sin^{13}(a) - 5602573729792\, \sin^{11}(a) + 124250627072\, \sin^{9}(a) - 1792076352\, \sin^{7}(a) + 15005424\, \sin^{5}(a) - 59640\, \sin^{3}(a) + 71\, \sin(a)$ $\sin(72a) = \big( - 2361183241434822606848\, \sin^{71}(a) + 41320706725109395619840\, \sin^{69}(a) - 346208492775380864729088\, \sin^{67}(a) + 1848954051996055777574912\, \sin^{65}(a) - 7069530198808448561315840\, \sin^{63}(a) + 20607152952646119463059456\, \sin^{61}(a) - 47615012504220200274493440\, \sin^{59}(a) + 89526688345846991944679424\, \sin^{57}(a) - 139535736914034960101277696\, \sin^{55}(a) + 182725369768379114418339840\, \sin^{53}(a) - 203060935113569693281026048\, \sin^{51}(a) + 192923019575112786090393600\, \sin^{49}(a) - 157553799319675441973821440\, \sin^{47}(a) + 111027155843917309500456960\, \sin^{45}(a) - 67682810520614616013209600\, \sin^{43}(a) + 35741273625798244596449280\, \sin^{41}(a) - 16354823868947857460428800\, \sin^{39}(a) + 6480708281759552073891840\, \sin^{37}(a) - 2220242652084290988277760\, \sin^{35}(a) + 655930674275150515404800\, \sin^{33}(a) - 166505478854461284679680\, \sin^{31}(a) + 36146147370366245273600\, \sin^{29}(a) - 6670607196531225264128\, \sin^{27}(a) + 1038768023949627359232\, \sin^{25}(a) - 135256253118441062400\, \sin^{23}(a) + 14561630654878973952\, \sin^{21}(a) - 1278404029065461760\, \sin^{19}(a) + 89961765008310272\, \sin^{17}(a) - 4965422094614528\, \sin^{15}(a) + 209049446645760\, \sin^{13}(a) - 6470578110464\, \sin^{11}(a) + 140000706560\, \sin^{9}(a) - 1968759936\, \sin^{7}(a) + 16062144\, \sin^{5}(a) - 62160\, \sin^{3}(a) + 72\, \sin(a)\big) \cdot \cos(a)$ $\sin(73a) = 4722366482869645213696\, \sin^{73}(a) - 86183188312371025149952\, \sin^{71}(a) + 754102897733246470062080\, \sin^{69}(a) - 4212203328767133854203904\, \sin^{67}(a) + 16871705724464008970371072\, \sin^{65}(a) - 51607570451301674497605632\, \sin^{63}(a) + 125360180461930560066945024\, \sin^{61}(a) - 248278279486291044288430080\, \sin^{59}(a) + 408465515577926900747599872\, \sin^{57}(a) - 565894933040252893744070656\, \sin^{55}(a) + 666947599654583767626940416\, \sin^{53}(a) - 673793102876844891341586432\, \sin^{51}(a) + 586807517874301391024947200\, \sin^{49}(a) - 442362590397550279388037120\, \sin^{47}(a) + 289463656307355842626191360\, \sin^{45}(a) - 164694838933495565632143360\, \sin^{43}(a) + 81534780458852245485649920\, \sin^{41}(a) - 35114768895093929253273600\, \sin^{39}(a) + 13141436238012425038725120\, \sin^{37}(a) - 4265202989530348477480960\, \sin^{35}(a) + 1197073480552149690613760\, \sin^{33}(a) - 289402379913706518609920\, \sin^{31}(a) + 59969744500834906931200\, \sin^{29}(a) - 10585963594495205310464\, \sin^{27}(a) + 1579793036423391608832\, \sin^{25}(a) - 197474129552923951104\, \sin^{23}(a) + 20442289188580098048\, \sin^{21}(a) - 1728212854107013120\, \sin^{19}(a) + 117271586528690176\, \sin^{17}(a) - 6249582981152768\, \sin^{15}(a) + 254343493419008\, \sin^{13}(a) - 7618583904256\, \sin^{11}(a) + 159688305920\, \sin^{9}(a) - 2177567808\, \sin^{7}(a) + 17243184\, \sin^{5}(a) - 64824\, \sin^{3}(a) + 73\, \sin(a)$ $\sin(74a) = \big(9444732965739290427392\, \sin^{73}(a) - 170005193383307227693056\, \sin^{71}(a) + 1466885088741383544504320\, \sin^{69}(a) - 8078198164758886843678720\, \sin^{67}(a) + 31894457396931962163167232\, \sin^{65}(a) - 96145610703794900433895424\, \sin^{63}(a) + 230113207971215000670830592\, \sin^{61}(a) - 448941546468361888302366720\, \sin^{59}(a) + 727404342810006809550520320\, \sin^{57}(a) - 992254129166470827386863616\, \sin^{55}(a) + 1151169829540788420835540992\, \sin^{53}(a) - 1144525270640120089402146816\, \sin^{51}(a) + 980692016173489995959500800\, \sin^{49}(a) - 727171381475425116802252800\, \sin^{47}(a) + 467900156770794375751925760\, \sin^{45}(a) - 261706867346376515251077120\, \sin^{43}(a) + 127328287291906246374850560\, \sin^{41}(a) - 53874713921240001046118400\, \sin^{39}(a) + 19802164194265298003558400\, \sin^{37}(a) - 6310163326976405966684160\, \sin^{35}(a) + 1738216286829148865822720\, \sin^{33}(a) - 412299280972951752540160\, \sin^{31}(a) + 83793341631303568588800\, \sin^{29}(a) - 14501319992459185356800\, \sin^{27}(a) + 2120818048897155858432\, \sin^{25}(a) - 259692005987406839808\, \sin^{23}(a) + 26322947722281222144\, \sin^{21}(a) - 2178021679148564480\, \sin^{19}(a) + 144581408049070080\, \sin^{17}(a) - 7533743867691008\, \sin^{15}(a) + 299637540192256\, \sin^{13}(a) - 8766589698048\, \sin^{11}(a) + 179375905280\, \sin^{9}(a) - 2386375680\, \sin^{7}(a) + 18424224\, \sin^{5}(a) - 67488\, \sin^{3}(a) + 74\, \sin(a)\big) \cdot \cos(a)$ $\sin(75a) = - 18889465931478580854784\, \sin^{75}(a) + 354177486215223391027200\, \sin^{73}(a) - 3187597375937010519244800\, \sin^{71}(a) + 18336063609267294306304000\, \sin^{69}(a) - 75733107794614564159488000\, \sin^{67}(a) + 239208430476989716223754240\, \sin^{65}(a) - 600910066898718127711846400\, \sin^{63}(a) + 1232749328417223217879449600\, \sin^{61}(a) - 2104413499070446351417344000\, \sin^{59}(a) + 3030851428375028373127168000\, \sin^{57}(a) - 3720952984374265602700738560\, \sin^{55}(a) + 3924442600707233252848435200\, \sin^{53}(a) - 3576641470750375279381708800\, \sin^{51}(a) + 2828919277423528834498560000\, \sin^{49}(a) - 1947780486094888705720320000\, \sin^{47}(a) + 1169750391926985939379814400\, \sin^{45}(a) - 613375470343069957619712000\, \sin^{43}(a) + 280871221967440249356288000\, \sin^{41}(a) - 112238987335916668846080000\, \sin^{39}(a) + 39083218804470982901760000\, \sin^{37}(a) - 11831556238080761187532800\, \sin^{35}(a) + 3103957655052051546112000\, \sin^{33}(a) - 702782865294804123648000\, \sin^{31}(a) + 136619578746690600960000\, \sin^{29}(a) - 22658312488217477120000\, \sin^{27}(a) + 3181227073345733787648\, \sin^{25}(a) - 374555777866452172800\, \sin^{23}(a) + 36559649614279475200\, \sin^{21}(a) - 2916993320288256000\, \sin^{19}(a) + 186958717304832000\, \sin^{17}(a) - 9417179834613760\, \sin^{15}(a) + 362464766361600\, \sin^{13}(a) - 10273347302400\, \sin^{11}(a) + 203836256000\, \sin^{9}(a) - 2632032000\, \sin^{7}(a) + 19740240\, \sin^{5}(a) - 70300\, \sin^{3}(a) + 75\, \sin(a)$ $\sin(76a) = \big( - 37778931862957161709568\, \sin^{75}(a) + 698910239464707491627008\, \sin^{73}(a) - 6205189558490713810796544\, \sin^{71}(a) + 35205242129793205068103680\, \sin^{69}(a) - 143388017424470241475297280\, \sin^{67}(a) + 446522403557047470284341248\, \sin^{65}(a) - 1105674523093641354989797376\, \sin^{63}(a) + 2235385448863231435088068608\, \sin^{61}(a) - 3759885451672530814532321280\, \sin^{59}(a) + 5334298513940049936703815680\, \sin^{57}(a) - 6449651839582060378014613504\, \sin^{55}(a) + 6697715371873678084861329408\, \sin^{53}(a) - 6008757670860630469361270784\, \sin^{51}(a) + 4677146538673567673037619200\, \sin^{49}(a) - 3168389590714352294638387200\, \sin^{47}(a) + 1871600627083177503007703040\, \sin^{45}(a) - 965044073339763399988346880\, \sin^{43}(a) + 434414156642974252337725440\, \sin^{41}(a) - 170603260750593336646041600\, \sin^{39}(a) + 58364273414676667799961600\, \sin^{37}(a) - 17352949149185116408381440\, \sin^{35}(a) + 4469699023274954226401280\, \sin^{33}(a) - 993266449616656494755840\, \sin^{31}(a) + 189445815862077633331200\, \sin^{29}(a) - 30815304983975768883200\, \sin^{27}(a) + 4241636097794311716864\, \sin^{25}(a) - 489419549745497505792\, \sin^{23}(a) + 46796351506277728256\, \sin^{21}(a) - 3655964961427947520\, \sin^{19}(a) + 229336026560593920\, \sin^{17}(a) - 11300615801536512\, \sin^{15}(a) + 425291992530944\, \sin^{13}(a) - 11780104906752\, \sin^{11}(a) + 228296606720\, \sin^{9}(a) - 2877688320\, \sin^{7}(a) + 21056256\, \sin^{5}(a) - 73112\, \sin^{3}(a) + 76\, \sin(a)\big) \cdot \cos(a)$ $\sin(77a) = 75557863725914323419136\, \sin^{77}(a) - 1454488876723850725818368\, \sin^{75}(a) + 13454022109695619213819904\, \sin^{73}(a) - 79633266000630827238555648\, \sin^{71}(a) + 338850455499259598780497920\, \sin^{69}(a) - 1104087734168420859359789056\, \sin^{67}(a) + 2865185422824387934324523008\, \sin^{65}(a) - 6081209877015027452443885568\, \sin^{63}(a) + 10757792472654301281361330176\, \sin^{61}(a) - 16083954432154715151054929920\, \sin^{59}(a) + 20537049278669192256309690368\, \sin^{57}(a) - 22573781438537211323051147264\, \sin^{55}(a) + 21488503484761383855596765184\, \sin^{53}(a) - 17795166948318021005416071168\, \sin^{51}(a) + 12862152981352311100853452800\, \sin^{49}(a) - 8132199949500170889571860480\, \sin^{47}(a) + 4503539008918895866612285440\, \sin^{45}(a) - 2185540989622405347032432640\, \sin^{43}(a) + 929163612819694928611246080\, \sin^{41}(a) - 345696080994623340045926400\, \sin^{39}(a) + 112351226323252585514926080\, \sin^{37}(a) - 31813740106839380082032640\, \sin^{35}(a) + 7821973290731169896202240\, \sin^{33}(a) - 1662641665662664132526080\, \sin^{31}(a) + 303902662945416203468800\, \sin^{29}(a) - 47455569675322684080128\, \sin^{27}(a) + 6280884221733884657664\, \sin^{25}(a) - 697876024637098295296\, \sin^{23}(a) + 64344983321131876352\, \sin^{21}(a) - 4853608655688826880\, \sin^{19}(a) + 294314567419428864\, \sin^{17}(a) - 14034635753521152\, \sin^{15}(a) + 511679428513792\, \sin^{13}(a) - 13743455724544\, \sin^{11}(a) + 258512334080\, \sin^{9}(a) - 3165457152\, \sin^{7}(a) + 22518496\, \sin^{5}(a) - 76076\, \sin^{3}(a) + 77\, \sin(a)$ $\sin(78a) = \big(151115727451828646838272\, \sin^{77}(a) - 2871198821584744289927168\, \sin^{75}(a) + 26209133979926530936012800\, \sin^{73}(a) - 153061342442770940666314752\, \sin^{71}(a) + 642495668868725992492892160\, \sin^{69}(a) - 2064787450912371477244280832\, \sin^{67}(a) + 5283848442091728398364704768\, \sin^{65}(a) - 11056745230936413549897973760\, \sin^{63}(a) + 19280199496445371127634591744\, \sin^{61}(a) - 28408023412636899487577538560\, \sin^{59}(a) + 35739800043398334575915565056\, \sin^{57}(a) - 38697911037492362268087681024\, \sin^{55}(a) + 36279291597649089626332200960\, \sin^{53}(a) - 29581576225775411541470871552\, \sin^{51}(a) + 21047159424031054528669286400\, \sin^{49}(a) - 13096010308285989484505333760\, \sin^{47}(a) + 7135477390754614230216867840\, \sin^{45}(a) - 3406037905905047294076518400\, \sin^{43}(a) + 1423913068996415604884766720\, \sin^{41}(a) - 520788901238653343445811200\, \sin^{39}(a) + 166338179231828503229890560\, \sin^{37}(a) - 46274531064493643755683840\, \sin^{35}(a) + 11174247558187385566003200\, \sin^{33}(a) - 2332016881708671770296320\, \sin^{31}(a) + 418359510028754773606400\, \sin^{29}(a) - 64095834366669599277056\, \sin^{27}(a) + 8320132345673457598464\, \sin^{25}(a) - 906332499528699084800\, \sin^{23}(a) + 81893615135986024448\, \sin^{21}(a) - 6051252349949706240\, \sin^{19}(a) + 359293108278263808\, \sin^{17}(a) - 16768655705505792\, \sin^{15}(a) + 598066864496640\, \sin^{13}(a) - 15706806542336\, \sin^{11}(a) + 288728061440\, \sin^{9}(a) - 3453225984\, \sin^{7}(a) + 23980736\, \sin^{5}(a) - 79040\, \sin^{3}(a) + 78\, \sin(a)\big) \cdot \cos(a)$ $\sin(79a) = - 302231454903657293676544\, \sin^{79}(a) + 5969071234347231550111744\, \sin^{77}(a) - 56706176726298699726061568\, \sin^{75}(a) + 345086930735699323990835200\, \sin^{73}(a) - 1511480756622363039079858176\, \sin^{71}(a) + 5075715784062935340693848064\, \sin^{69}(a) - 13593184051839778891858182144\, \sin^{67}(a) + 29816001923231895962200834048\, \sin^{65}(a) - 54592679577748541902621245440\, \sin^{63}(a) + 84618653345510239949062930432\, \sin^{61}(a) - 112211692479915752975931277312\, \sin^{59}(a) + 128338372883112201431696801792\, \sin^{57}(a) - 127380623831745692465788616704\, \sin^{55}(a) + 110233232162087618480009379840\, \sin^{53}(a) - 83462304351294911134864244736\, \sin^{51}(a) + 55424186483281776925495787520\, \sin^{49}(a) - 32330775448581036539872542720\, \sin^{47}(a) + 16579491584400427181974487040\, \sin^{45}(a) - 7474360960180520450890137600\, \sin^{43}(a) + 2960240327650442968049909760\, \sin^{41}(a) - 1028558079946340353305477120\, \sin^{39}(a) + 312874194269391708456222720\, \sin^{37}(a) - 83083817138522678561341440\, \sin^{35}(a) + 19190555589060944776396800\, \sin^{33}(a) - 3838111117812188955279360\, \sin^{31}(a) + 661008025845432542298112\, \sin^{29}(a) - 97376363749363429670912\, \sin^{27}(a) + 12172045468670428708864\, \sin^{25}(a) - 1278576204692271923200\, \sin^{23}(a) + 111544751650739585024\, \sin^{21}(a) - 7967482260767113216\, \sin^{19}(a) + 457808960548110336\, \sin^{17}(a) - 20698809386483712\, \sin^{15}(a) + 715867913564160\, \sin^{13}(a) - 18247613483008\, \sin^{11}(a) + 325850240768\, \sin^{9}(a) - 3788956288\, \sin^{7}(a) + 25601056\, \sin^{5}(a) - 82160\, \sin^{3}(a) + 79\, \sin(a)$ $\sin(80a) = \big( - 604462909807314587353088\, \sin^{79}(a) + 11787026741242634453385216\, \sin^{77}(a) - 110541154631012655162195968\, \sin^{75}(a) + 663964727491472117045657600\, \sin^{73}(a) - 2869900170801955137493401600\, \sin^{71}(a) + 9508935899257144688894803968\, \sin^{69}(a) - 25121580652767186306472083456\, \sin^{67}(a) + 54348155404372063526036963328\, \sin^{65}(a) - 98128613924560670255344517120\, \sin^{63}(a) + 149957107194575108770491269120\, \sin^{61}(a) - 196015361547194606464285016064\, \sin^{59}(a) + 220936945722826068287478038528\, \sin^{57}(a) - 216063336625999022663489552384\, \sin^{55}(a) + 184187172726526147333686558720\, \sin^{53}(a) - 137343032476814410728257617920\, \sin^{51}(a) + 89801213542532499322322288640\, \sin^{49}(a) - 51565540588876083595239751680\, \sin^{47}(a) + 26023505778046240133732106240\, \sin^{45}(a) - 11542684014455993607703756800\, \sin^{43}(a) + 4496567586304470331215052800\, \sin^{41}(a) - 1536327258654027363165143040\, \sin^{39}(a) + 459410209306954913682554880\, \sin^{37}(a) - 119893103212551713366999040\, \sin^{35}(a) + 27206863619934503986790400\, \sin^{33}(a) - 5344205353915706140262400\, \sin^{31}(a) + 903656541662110310989824\, \sin^{29}(a) - 130656893132057260064768\, \sin^{27}(a) + 16023958591667399819264\, \sin^{25}(a) - 1650819909855844761600\, \sin^{23}(a) + 141195888165493145600\, \sin^{21}(a) - 9883712171584520192\, \sin^{19}(a) + 556324812817956864\, \sin^{17}(a) - 24628963067461632\, \sin^{15}(a) + 833668962631680\, \sin^{13}(a) - 20788420423680\, \sin^{11}(a) + 362972420096\, \sin^{9}(a) - 4124686592\, \sin^{7}(a) + 27221376\, \sin^{5}(a) - 85280\, \sin^{3}(a) + 80\, \sin(a)\big) \cdot \cos(a)$ $\sin(81a) = 1208925819614629174706176\, \sin^{81}(a) - 24480747847196240787800064\, \sin^{79}(a) + 238687291510163347681050624\, \sin^{77}(a) - 1492305587518670844689645568\, \sin^{75}(a) + 6722642865851155185087283200\, \sin^{73}(a) - 23246191383495836613696552960\, \sin^{71}(a) + 64185317319985726650039926784\, \sin^{69}(a) - 145346288062438720773159911424\, \sin^{67}(a) + 275137536734633571600562126848\, \sin^{65}(a) - 441578762660523016149050327040\, \sin^{63}(a) + 607326284138029190520489639936\, \sin^{61}(a) - 721692922060125596527594831872\, \sin^{59}(a) + 745662191814537980470238380032\, \sin^{57}(a) - 673120394873304647528563605504\, \sin^{55}(a) + 532827178244593497643878973440\, \sin^{53}(a) - 370826187687398908966295568384\, \sin^{51}(a) + 227309321779535388909628293120\, \sin^{49}(a) - 122847317285263610918071173120\, \sin^{47}(a) + 58552888000604040300897239040\, \sin^{45}(a) - 24604142241340407426947481600\, \sin^{43}(a) + 9105549362266552420710481920\, \sin^{41}(a) - 2962916855975624200389918720\, \sin^{39}(a) + 845732430769621545642885120\, \sin^{37}(a) - 211116116526449756146237440\, \sin^{35}(a) + 45911582358639475477708800\, \sin^{33}(a) - 8657612673343443947225088\, \sin^{31}(a) + 1407618843742902599811072\, \sin^{29}(a) - 195985339698085890097152\, \sin^{27}(a) + 23177511534376060452864\, \sin^{25}(a) - 2305455391350403891200\, \sin^{23}(a) + 190614449023415746560\, \sin^{21}(a) - 12912591708037840896\, \sin^{19}(a) + 704098591222726656\, \sin^{17}(a) - 30226454673702912\, \sin^{15}(a) + 993046852546560\, \sin^{13}(a) - 24055172204544\, \sin^{11}(a) + 408343972608\, \sin^{9}(a) - 4514859648\, \sin^{7}(a) + 29012256\, \sin^{5}(a) - 88560\, \sin^{3}(a) + 81\, \sin(a)$ $\sin(82a) = \big(2417851639229258349412352\, \sin^{81}(a) - 48357032784585166988247040\, \sin^{79}(a) + 465587556279084060908716032\, \sin^{77}(a) - 2874070020406329034217095168\, \sin^{75}(a) + 12781321004210838253128908800\, \sin^{73}(a) - 43622482596189718089899704320\, \sin^{71}(a) + 118861698740714308611185049600\, \sin^{69}(a) - 265570995472110255239847739392\, \sin^{67}(a) + 495926918064895079675087290368\, \sin^{65}(a) - 785028911396485362042756136960\, \sin^{63}(a) + 1064695461081483272270488010752\, \sin^{61}(a) - 1247370482573056586590904647680\, \sin^{59}(a) + 1270387437906249892652998721536\, \sin^{57}(a) - 1130177453120610272393637658624\, \sin^{55}(a) + 881467183762660847954071388160\, \sin^{53}(a) - 604309342897983407204333518848\, \sin^{51}(a) + 364817430016538278496934297600\, \sin^{49}(a) - 194129093981651138240902594560\, \sin^{47}(a) + 91082270223161840468062371840\, \sin^{45}(a) - 37665600468224821246191206400\, \sin^{43}(a) + 13714531138228634510205911040\, \sin^{41}(a) - 4389506453297221037614694400\, \sin^{39}(a) + 1232054652232288177603215360\, \sin^{37}(a) - 302339129840347798925475840\, \sin^{35}(a) + 64616301097344446968627200\, \sin^{33}(a) - 11971019992771181754187776\, \sin^{31}(a) + 1911581145823694888632320\, \sin^{29}(a) - 261313786264114520129536\, \sin^{27}(a) + 30331064477084721086464\, \sin^{25}(a) - 2960090872844963020800\, \sin^{23}(a) + 240033009881338347520\, \sin^{21}(a) - 15941471244491161600\, \sin^{19}(a) + 851872369627496448\, \sin^{17}(a) - 35823946279944192\, \sin^{15}(a) + 1152424742461440\, \sin^{13}(a) - 27321923985408\, \sin^{11}(a) + 453715525120\, \sin^{9}(a) - 4905032704\, \sin^{7}(a) + 30803136\, \sin^{5}(a) - 91840\, \sin^{3}(a) + 82\, \sin(a)\big) \cdot \cos(a)$ $\sin(83a) = - 4835703278458516698824704\, \sin^{83}(a) + 100340843028014221500612608\, \sin^{81}(a) - 1003408430280142215006126080\, \sin^{79}(a) + 6440627861860662842570571776\, \sin^{77}(a) - 29818476461715663730002362368\, \sin^{75}(a) + 106084964334949957500969943040\, \sin^{73}(a) - 301722171290312216788472954880\, \sin^{71}(a) + 704680071105663401052025651200\, \sin^{69}(a) - 1377649539011571949056710148096\, \sin^{67}(a) + 2286774122188127311835124727808\, \sin^{65}(a) - 3257869982295414252477437968384\, \sin^{63}(a) + 4016805603171050527202295676928\, \sin^{61}(a) - 4313822918898487361960211906560\, \sin^{59}(a) + 4055467590239182349623034380288\, \sin^{57}(a) - 3350168878893237593166854488064\, \sin^{55}(a) + 2438725875076695012672930840576\, \sin^{53}(a) - 1567427358141644462436240064512\, \sin^{51}(a) + 890583726216843444566045491200\, \sin^{49}(a) - 447575411124362346499858759680\, \sin^{47}(a) + 198942853382169283127609917440\, \sin^{45}(a) - 78156120971566504085846753280\, \sin^{43}(a) + 27102525820785158674930728960\, \sin^{41}(a) - 8280205355083394230045900800\, \sin^{39}(a) + 2223055133375650407414497280\, \sin^{37}(a) - 522794745348934735641968640\, \sin^{35}(a) + 107263059821591781967921152\, \sin^{33}(a) - 19107589603846309338415104\, \sin^{31}(a) + 2938171020432716217712640\, \sin^{29}(a) - 387304361784312592334848\, \sin^{27}(a) + 43404799165483307761664\, \sin^{25}(a) - 4094792374102198845440\, \sin^{23}(a) + 321334513228243271680\, \sin^{21}(a) - 20674095520199475200\, \sin^{19}(a) + 1071294040592154624\, \sin^{17}(a) - 43726287371108352\, \sin^{15}(a) + 1366446480347136\, \sin^{13}(a) - 31496106816512\, \sin^{11}(a) + 508897143040\, \sin^{9}(a) - 5356812032\, \sin^{7}(a) + 32777696\, \sin^{5}(a) - 95284\, \sin^{3}(a) + 83\, \sin(a)$ $\sin(84a) = \big( - 9671406556917033397649408\, \sin^{83}(a) + 198263834416799184651812864\, \sin^{81}(a) - 1958459827775699263024005120\, \sin^{79}(a) + 12415668167442241624232427520\, \sin^{77}(a) - 56762882903024998425787629568\, \sin^{75}(a) + 199388607665689076748810977280\, \sin^{73}(a) - 559821859984434715487046205440\, \sin^{71}(a) + 1290498443470612493492866252800\, \sin^{69}(a) - 2489728082551033642873572556800\, \sin^{67}(a) + 4077621326311359543995162165248\, \sin^{65}(a) - 5730711053194343142912119799808\, \sin^{63}(a) + 6968915745260617782134103343104\, \sin^{61}(a) - 7380275355223918137329519165440\, \sin^{59}(a) + 6840547742572114806593070039040\, \sin^{57}(a) - 5570160304665864913940071317504\, \sin^{55}(a) + 3995984566390729177391790292992\, \sin^{53}(a) - 2530545373385305517668146610176\, \sin^{51}(a) + 1416350022417148610635156684800\, \sin^{49}(a) - 701021728267073554758814924800\, \sin^{47}(a) + 306803436541176725787157463040\, \sin^{45}(a) - 118646641474908186925502300160\, \sin^{43}(a) + 40490520503341682839655546880\, \sin^{41}(a) - 12170904256869567422477107200\, \sin^{39}(a) + 3214055614519012637225779200\, \sin^{37}(a) - 743250360857521672358461440\, \sin^{35}(a) + 149909818545839116967215104\, \sin^{33}(a) - 26244159214921436922642432\, \sin^{31}(a) + 3964760895041737546792960\, \sin^{29}(a) - 513294937304510664540160\, \sin^{27}(a) + 56478533853881894436864\, \sin^{25}(a) - 5229493875359434670080\, \sin^{23}(a) + 402636016575148195840\, \sin^{21}(a) - 25406719795907788800\, \sin^{19}(a) + 1290715711556812800\, \sin^{17}(a) - 51628628462272512\, \sin^{15}(a) + 1580468218232832\, \sin^{13}(a) - 35670289647616\, \sin^{11}(a) + 564078760960\, \sin^{9}(a) - 5808591360\, \sin^{7}(a) + 34752256\, \sin^{5}(a) - 98728\, \sin^{3}(a) + 84\, \sin(a)\big) \cdot \cos(a)$ $\sin(85a) = 19342813113834066795298816\, \sin^{85}(a) - 411034778668973919400099840\, \sin^{83}(a) + 4213106481356982673851023360\, \sin^{81}(a) - 27744847560155739559506739200\, \sin^{79}(a) + 131916474279073817257469542400\, \sin^{77}(a) - 482484504675712486619194851328\, \sin^{75}(a) + 1412335970965297626970744422400\, \sin^{73}(a) - 3398918435619782201171351961600\, \sin^{71}(a) + 6855772980937628871680851968000\, \sin^{69}(a) - 11757049278713214424680759296000\, \sin^{67}(a) + 17329890636823278061979439202304\, \sin^{65}(a) - 22141383614614507597615008317440\, \sin^{63}(a) + 24681576597798021311724949340160\, \sin^{61}(a) - 24127823276693578525884966502400\, \sin^{59}(a) + 20765948504236777091443248332800\, \sin^{57}(a) - 15782120863219950589496868732928\, \sin^{55}(a) + 10614334004475374377446942965760\, \sin^{53}(a) - 6326363433463263794170366525440\, \sin^{51}(a) + 3344159775151600886221897728000\, \sin^{49}(a) - 1568074918492138214592086016000\, \sin^{47}(a) + 651957302650000542297709608960\, \sin^{45}(a) - 240118202984933235444468940800\, \sin^{43}(a) + 78220323699637341849334579200\, \sin^{41}(a) - 22489714387693765889359872000\, \sin^{39}(a) + 5691556817377418211753984000\, \sin^{37}(a) - 1263525613457786843009384448\, \sin^{35}(a) + 245044895699929325811793920\, \sin^{33}(a) - 41310250616080039600455680\, \sin^{31}(a) + 6017940644259780204953600\, \sin^{29}(a) - 752242580532472525619200\, \sin^{27}(a) + 80011256292999350452224\, \sin^{25}(a) - 7169467409766966886400\, \sin^{23}(a) + 534750959513868697600\, \sin^{21}(a) - 32720775494729728000\, \sin^{19}(a) + 1613394639446016000\, \sin^{17}(a) - 62691905989902336\, \sin^{15}(a) + 1865830535413760\, \sin^{13}(a) - 40972630000640\, \sin^{11}(a) + 630877561600\, \sin^{9}(a) - 6329875200\, \sin^{7}(a) + 36924272\, \sin^{5}(a) - 102340\, \sin^{3}(a) + 85\, \sin(a)$ $\sin(86a) = \big(38685626227668133590597632\, \sin^{85}(a) - 812398150781030805402550272\, \sin^{83}(a) + 8227949128297166163050233856\, \sin^{81}(a) - 53531235292535779855989473280\, \sin^{79}(a) + 251417280390705392890706657280\, \sin^{77}(a) - 908206126448399974812602073088\, \sin^{75}(a) + 2625283334264906177192677867520\, \sin^{73}(a) - 6238015011255129686855657717760\, \sin^{71}(a) + 12421047518404645249868837683200\, \sin^{69}(a) - 21024370474875395206487946035200\, \sin^{67}(a) + 30582159947335196579963716239360\, \sin^{65}(a) - 38552056176034672052317896835072\, \sin^{63}(a) + 42394237450335424841315795337216\, \sin^{61}(a) - 40875371198163238914440413839360\, \sin^{59}(a) + 34691349265901439376293426626560\, \sin^{57}(a) - 25994081421774036265053666148352\, \sin^{55}(a) + 17232683442560019577502095638528\, \sin^{53}(a) - 10122181493541222070672586440704\, \sin^{51}(a) + 5271969527886053161808638771200\, \sin^{49}(a) - 2435128108717202874425357107200\, \sin^{47}(a) + 997111168758824358808261754880\, \sin^{45}(a) - 361589764494958283963435581440\, \sin^{43}(a) + 115950126895933000859013611520\, \sin^{41}(a) - 32808524518517964356242636800\, \sin^{39}(a) + 8169058020235823786282188800\, \sin^{37}(a) - 1783800866058052013660307456\, \sin^{35}(a) + 340179972854019534656372736\, \sin^{33}(a) - 56376342017238642278268928\, \sin^{31}(a) + 8071120393477822863114240\, \sin^{29}(a) - 991190223760434386698240\, \sin^{27}(a) + 103543978732116806467584\, \sin^{25}(a) - 9109440944174499102720\, \sin^{23}(a) + 666865902452589199360\, \sin^{21}(a) - 40034831193551667200\, \sin^{19}(a) + 1936073567335219200\, \sin^{17}(a) - 73755183517532160\, \sin^{15}(a) + 2151192852594688\, \sin^{13}(a) - 46274970353664\, \sin^{11}(a) + 697676362240\, \sin^{9}(a) - 6851159040\, \sin^{7}(a) + 39096288\, \sin^{5}(a) - 105952\, \sin^{3}(a) + 86\, \sin(a)\big) \cdot \cos(a)$ $\sin(87a) = - 77371252455336267181195264\, \sin^{87}(a) + 1682824740903563811190996992\, \sin^{85}(a) - 17669659779487420017505468416\, \sin^{83}(a) + 119305262360308909364228390912\, \sin^{81}(a) - 582152183806326605933885521920\, \sin^{79}(a) + 2187330339399136918149147918336\, \sin^{77}(a) - 6584494416750899817391365029888\, \sin^{75}(a) + 16314260720074774101125926748160\, \sin^{73}(a) - 33919206623699767672277638840320\, \sin^{71}(a) + 60035063005622452041032715468800\, \sin^{69}(a) - 91456011565707969148222565253120\, \sin^{67}(a) + 120938541609916459202583786946560\, \sin^{65}(a) - 139751203638125686189652376027136\, \sin^{63}(a) + 141857640699199306199787469012992\, \sin^{61}(a) - 127005617651435778055582714429440\, \sin^{59}(a) + 100604912871114174191250937217024\, \sin^{57}(a) - 70671408865448161095614654840832\, \sin^{55}(a) + 44095395867727108918902421192704\, \sin^{53}(a) - 24461938609391286670792083898368\, \sin^{51}(a) + 12070035498054911186246094028800\, \sin^{49}(a) - 5296403636459916251875151708160\, \sin^{47}(a) + 2065444563857564743245685063680\, \sin^{45}(a) - 714961579796849334200429445120\, \sin^{43}(a) + 219296979129264588581177917440\, \sin^{41}(a) - 59465450689813810395689779200\, \sin^{39}(a) + 14214160955210333388131008512\, \sin^{37}(a) - 2984436064366356253623975936\, \sin^{35}(a) + 548067734042587028057489408\, \sin^{33}(a) - 87584674205352890682310656\, \sin^{31}(a) + 12106680590216734294671360\, \sin^{29}(a) - 1437225824452629860712448\, \sin^{27}(a) + 145295583059583260688384\, \sin^{25}(a) - 12383146283487209717760\, \sin^{23}(a) + 879050507778413035520\, \sin^{21}(a) - 51221034027044044800\, \sin^{19}(a) + 2406262862259486720\, \sin^{17}(a) - 89120846750351360\, \sin^{15}(a) + 2529105110482944\, \sin^{13}(a) - 52972663431168\, \sin^{11}(a) + 778177480960\, \sin^{9}(a) - 7450635456\, \sin^{7}(a) + 41480208\, \sin^{5}(a) - 109736\, \sin^{3}(a) + 87\, \sin(a)$ $\sin(88a) = \big( - 154742504910672534362390528\, \sin^{87}(a) + 3326963855579459488791396352\, \sin^{85}(a) - 34526921408193809229608386560\, \sin^{83}(a) + 230382575592320652565406547968\, \sin^{81}(a) - 1110773132320117432011781570560\, \sin^{79}(a) + 4123243398407568443407589179392\, \sin^{77}(a) - 12260782707053399659970127986688\, \sin^{75}(a) + 30003238105884642025059175628800\, \sin^{73}(a) - 61600398236144405657699619962880\, \sin^{71}(a) + 107649078492840258832196593254400\, \sin^{69}(a) - 161887652656540543089957184471040\, \sin^{67}(a) + 211294923272497721825203857653760\, \sin^{65}(a) - 240950351100216700326986855219200\, \sin^{63}(a) + 241321043948063187558259142688768\, \sin^{61}(a) - 213135864104708317196725015019520\, \sin^{59}(a) + 166518476476326909006208447807488\, \sin^{57}(a) - 115348736309122285926175643533312\, \sin^{55}(a) + 70958108292894198260302746746880\, \sin^{53}(a) - 38801695725241351270911581356032\, \sin^{51}(a) + 18868101468223769210683549286400\, \sin^{49}(a) - 8157679164202629629324946309120\, \sin^{47}(a) + 3133777958956305127683108372480\, \sin^{45}(a) - 1068333395098740384437423308800\, \sin^{43}(a) + 322643831362596176303342223360\, \sin^{41}(a) - 86122376861109656435136921600\, \sin^{39}(a) + 20259263890184842989979828224\, \sin^{37}(a) - 4185071262674660493587644416\, \sin^{35}(a) + 755955495231154521458606080\, \sin^{33}(a) - 118793006393467139086352384\, \sin^{31}(a) + 16142240786955645726228480\, \sin^{29}(a) - 1883261425144825334726656\, \sin^{27}(a) + 187047187387049714909184\, \sin^{25}(a) - 15656851622799920332800\, \sin^{23}(a) + 1091235113104236871680\, \sin^{21}(a) - 62407236860536422400\, \sin^{19}(a) + 2876452157183754240\, \sin^{17}(a) - 104486509983170560\, \sin^{15}(a) + 2907017368371200\, \sin^{13}(a) - 59670356508672\, \sin^{11}(a) + 858678599680\, \sin^{9}(a) - 8050111872\, \sin^{7}(a) + 43864128\, \sin^{5}(a) - 113520\, \sin^{3}(a) + 88\, \sin(a)\big) \cdot \cos(a)$ $\sin(89a) = 309485009821345068724781056\, \sin^{89}(a) - 6886041468524927779126378496\, \sin^{87}(a) + 74024945786642973625608568832\, \sin^{85}(a) - 512149334221541503572524400640\, \sin^{83}(a) + 2563006153464567259790147846144\, \sin^{81}(a) - 9885880877649045144904855977984\, \sin^{79}(a) + 30580721871522799288606286413824\, \sin^{77}(a) - 77943547209125183552667242201088\, \sin^{75}(a) + 166893011963983321264391664435200\, \sin^{73}(a) - 304579746834269561307514787594240\, \sin^{71}(a) + 479038399293139151803274839982080\, \sin^{69}(a) - 654909140292368560682099518996480\, \sin^{67}(a) + 783552007135512385101797638799360\, \sin^{65}(a) - 824791586458434089580839619788800\, \sin^{63}(a) + 767056175406343703310180846403584\, \sin^{61}(a) - 632303063510634674350284211224576\, \sin^{59}(a) + 463129512699784215673517245464576\, \sin^{57}(a) - 301942280338584807277342125719552\, \sin^{55}(a) + 175424212168543990143526235013120\, \sin^{53}(a) - 90877655777538954292398177386496\, \sin^{51}(a) + 41981525766797886493770897162240\, \sin^{49}(a) - 17286510609857953262140957655040\, \sin^{47}(a) + 6338778144252526280995378298880\, \sin^{45}(a) - 2066992873125823787281101619200\, \sin^{43}(a) + 598235437318147076895780372480\, \sin^{41}(a) - 153297830812775188454543720448\, \sin^{39}(a) + 34674509350508673579003936768\, \sin^{37}(a) - 6897617451445273776468525056\, \sin^{35}(a) + 1201429269206656293032427520\, \sin^{33}(a) - 182285820155492678942851072\, \sin^{31}(a) + 23944323833984207827238912\, \sin^{29}(a) - 2703391400611120238559232\, \sin^{27}(a) + 260112494960116009795584\, \sin^{25}(a) - 21113027188321104691200\, \sin^{23}(a) + 1428234192151133552640\, \sin^{21}(a) - 79346344008396308480\, \sin^{19}(a) + 3555614472074362880\, \sin^{17}(a) - 125666207952732160\, \sin^{15}(a) + 3404270339276800\, \sin^{13}(a) - 68085406785536\, \sin^{11}(a) + 955279942144\, \sin^{9}(a) - 8737316544\, \sin^{7}(a) + 46475088\, \sin^{5}(a) - 117480\, \sin^{3}(a) + 89\, \sin(a)$ $\sin(90a) = \big(618970019642690137449562112\, \sin^{89}(a) - 13617340432139183023890366464\, \sin^{87}(a) + 144722927717706487762425741312\, \sin^{85}(a) - 989771747034889197915440414720\, \sin^{83}(a) + 4895629731336813867014889144320\, \sin^{81}(a) - 18660988622977972857797930385408\, \sin^{79}(a) + 57038200344638030133804983648256\, \sin^{77}(a) - 143626311711196967445364356415488\, \sin^{75}(a) + 303782785822082000503724153241600\, \sin^{73}(a) - 547559095432394716957329955225600\, \sin^{71}(a) + 850427720093438044774353086709760\, \sin^{69}(a) - 1147930627928196578274241853521920\, \sin^{67}(a) + 1355809090998527048378391419944960\, \sin^{65}(a) - 1408632821816651478834692384358400\, \sin^{63}(a) + 1292791306864624219062102550118400\, \sin^{61}(a) - 1051470262916561031503843407429632\, \sin^{59}(a) + 759740548923241522340826043121664\, \sin^{57}(a) - 488535824368047328628508607905792\, \sin^{55}(a) + 279890316044193782026749723279360\, \sin^{53}(a) - 142953615829836557313884773416960\, \sin^{51}(a) + 65094950065372003776858245038080\, \sin^{49}(a) - 26415342055513276894956969000960\, \sin^{47}(a) + 9543778329548747434307648225280\, \sin^{45}(a) - 3065652351152907190124779929600\, \sin^{43}(a) + 873827043273697977488218521600\, \sin^{41}(a) - 220473284764440720473950519296\, \sin^{39}(a) + 49089754810832504168028045312\, \sin^{37}(a) - 9610163640215887059349405696\, \sin^{35}(a) + 1646903043182158064606248960\, \sin^{33}(a) - 245778633917518218799349760\, \sin^{31}(a) + 31746406881012769928249344\, \sin^{29}(a) - 3523521376077415142391808\, \sin^{27}(a) + 333177802533182304681984\, \sin^{25}(a) - 26569202753842289049600\, \sin^{23}(a) + 1765233271198030233600\, \sin^{21}(a) - 96285451156256194560\, \sin^{19}(a) + 4234776786964971520\, \sin^{17}(a) - 146845905922293760\, \sin^{15}(a) + 3901523310182400\, \sin^{13}(a) - 76500457062400\, \sin^{11}(a) + 1051881284608\, \sin^{9}(a) - 9424521216\, \sin^{7}(a) + 49086048\, \sin^{5}(a) - 121440\, \sin^{3}(a) + 90\, \sin(a)\big) \cdot \cos(a)$ $\sin(91a) = - 1237940039285380274899124224\, \sin^{91}(a) + 28163135893742401253955076096\, \sin^{89}(a) - 309794494831166413793505837056\, \sin^{87}(a) + 2194964403718548397730123743232\, \sin^{85}(a) - 11258653622521864626288134717440\, \sin^{83}(a) + 44550230555165006189835491213312\, \sin^{81}(a) - 141512497057582960838300972089344\, \sin^{79}(a) + 370748302240147195869732393713664\, \sin^{77}(a) - 816874647857432752345509777113088\, \sin^{75}(a) + 1535790750544970113657716552499200\, \sin^{73}(a) - 2491393884217395962155851296276480\, \sin^{71}(a) + 3517678296750130094293915040481280\, \sin^{69}(a) - 4352570297561078692623167027937280\, \sin^{67}(a) + 4745331818494844669324369969807360\, \sin^{65}(a) - 4578056670904117306212750249164800\, \sin^{63}(a) + 3921466964156026797821711068692480\, \sin^{61}(a) - 2990118560168970433339054689878016\, \sin^{59}(a) + 2033423233882793486265152056590336\, \sin^{57}(a) - 1234910000485897414033174536650752\, \sin^{55}(a) + 670263651579516688537742758379520\, \sin^{53}(a) - 325219476012878167889087859523584\, \sin^{51}(a) + 141039058474972674849859530915840\, \sin^{49}(a) - 54631730160266095396388276797440\, \sin^{47}(a) + 18880083217150782967869478010880\, \sin^{45}(a) - 5811965915727386547944895283200\, \sin^{43}(a) + 1590365218758130319028557709312\, \sin^{41}(a) - 385828248337771260829413408768\, \sin^{39}(a) + 82725327551588108875750965248\, \sin^{37}(a) - 15616515915350816471442784256\, \sin^{35}(a) + 2583934084992696273778769920\, \sin^{33}(a) - 372764261441569298512347136\, \sin^{31}(a) + 46595532680196162314043392\, \sin^{29}(a) - 5010006956610074655588352\, \sin^{27}(a) + 459381515613933177667584\, \sin^{25}(a) - 35555844861759533875200\, \sin^{23}(a) + 2294803252557439303680\, \sin^{21}(a) - 121694111878046023680\, \sin^{19}(a) + 5207630913700167680\, \sin^{17}(a) - 175828650512220160\, \sin^{15}(a) + 4551777195212800\, \sin^{13}(a) - 87019269908480\, \sin^{11}(a) + 1167331669504\, \sin^{9}(a) - 10209897984\, \sin^{7}(a) + 51939888\, \sin^{5}(a) - 125580\, \sin^{3}(a) + 91\, \sin(a)$ $\sin(92a) = \big( - 2475880078570760549798248448\, \sin^{91}(a) + 55707301767842112370460590080\, \sin^{89}(a) - 605971649230193644563121307648\, \sin^{87}(a) + 4245205879719390307697821745152\, \sin^{85}(a) - 21527535498008840054660829020160\, \sin^{83}(a) + 84204831378993198512656093282304\, \sin^{81}(a) - 264364005492187948818804013793280\, \sin^{79}(a) + 684458404135656361605659803779072\, \sin^{77}(a) - 1490122984003668537245655197810688\, \sin^{75}(a) + 2767798715267858226811708951756800\, \sin^{73}(a) - 4435228673002397207354372637327360\, \sin^{71}(a) + 6184928873406822143813476994252800\, \sin^{69}(a) - 7557209967193960806972092202352640\, \sin^{67}(a) + 8134854545991162290270348519669760\, \sin^{65}(a) - 7747480519991583133590808113971200\, \sin^{63}(a) + 6550142621447429376581319587266560\, \sin^{61}(a) - 4928766857421379835174265972326400\, \sin^{59}(a) + 3307105918842345450189478070059008\, \sin^{57}(a) - 1981284176603747499437840465395712\, \sin^{55}(a) + 1060636987114839595048735793479680\, \sin^{53}(a) - 507485336195919778464290945630208\, \sin^{51}(a) + 216983166884573345922860816793600\, \sin^{49}(a) - 82848118265018913897819584593920\, \sin^{47}(a) + 28216388104752818501431307796480\, \sin^{45}(a) - 8558279480301865905765010636800\, \sin^{43}(a) + 2306903394242562660568896897024\, \sin^{41}(a) - 551183211911101801184876298240\, \sin^{39}(a) + 116360900292343713583473885184\, \sin^{37}(a) - 21622868190485745883536162816\, \sin^{35}(a) + 3520965126803234482951290880\, \sin^{33}(a) - 499749888965620378225344512\, \sin^{31}(a) + 61444658479379554699837440\, \sin^{29}(a) - 6496492537142734168784896\, \sin^{27}(a) + 585585228694684050653184\, \sin^{25}(a) - 44542486969676778700800\, \sin^{23}(a) + 2824373233916848373760\, \sin^{21}(a) - 147102772599835852800\, \sin^{19}(a) + 6180485040435363840\, \sin^{17}(a) - 204811395102146560\, \sin^{15}(a) + 5202031080243200\, \sin^{13}(a) - 97538082754560\, \sin^{11}(a) + 1282782054400\, \sin^{9}(a) - 10995274752\, \sin^{7}(a) + 54793728\, \sin^{5}(a) - 129720\, \sin^{3}(a) + 92\, \sin(a)\big) \cdot \cos(a)$ $\sin(93a) = 4951760157141521099596496896\, \sin^{93}(a) - 115128423653540365565618552832\, \sin^{91}(a) + 1295194766102329112613208719360\, \sin^{89}(a) - 9392560563068001490728380268544\, \sin^{87}(a) + 49350518351737912326987177787392\, \sin^{85}(a) - 200206080131482212508345709887488\, \sin^{83}(a) + 652587443187197288473084722937856\, \sin^{81}(a) - 1756132322198105660010626663055360\, \sin^{79}(a) + 3978414474038502601832897609465856\, \sin^{77}(a) - 7698968750685620775769218522021888\, \sin^{75}(a) + 12870264025995540754674446625669120\, \sin^{73}(a) - 18748921208601042740179847966883840\, \sin^{71}(a) + 23966599384451435807277223352729600\, \sin^{69}(a) - 27031558728809167501861714416107520\, \sin^{67}(a) + 27019338313470646178397943297474560\, \sin^{65}(a) - 24017189611973907714131505153310720\, \sin^{63}(a) + 19036351993581591625689460050493440\, \sin^{61}(a) - 13481626992358480137388433394892800\, \sin^{59}(a) + 8543356957009392412989485014319104\, \sin^{57}(a) - 4848932326951276774939977981100032\, \sin^{55}(a) + 2465980995042002058488310719840256\, \sin^{53}(a) - 1123717530148108080885215665324032\, \sin^{51}(a) + 458623511824211844791501271859200\, \sin^{49}(a) - 167497282579277369402113507983360\, \sin^{47}(a) + 54669251952958585846523158855680\, \sin^{45}(a) - 15918399833361470584722919784448\, \sin^{43}(a) + 4125807993549198604478988681216\, \sin^{41}(a) - 949259976069119768707286958080\, \sin^{39}(a) + 193242209414070810058269130752\, \sin^{37}(a) - 34671150719227144261532123136\, \sin^{35}(a) + 5457495946545013448574500864\, \sin^{33}(a) - 749624833448430567338016768\, \sin^{31}(a) + 89286769352848415423201280\, \sin^{29}(a) - 9154148575064761783287808\, \sin^{27}(a) + 800873915714788481040384\, \sin^{25}(a) - 59177875545427720273920\, \sin^{23}(a) + 3648148760475929149440\, \sin^{21}(a) - 184872403402496409600\, \sin^{19}(a) + 7562961957374853120\, \sin^{17}(a) - 244198201852559360\, \sin^{15}(a) + 6047361130782720\, \sin^{13}(a) - 110622459709440\, \sin^{11}(a) + 1420222988800\, \sin^{9}(a) - 11890238976\, \sin^{7}(a) + 57907008\, \sin^{5}(a) - 134044\, \sin^{3}(a) + 93\, \sin(a)$ $\sin(94a) = \big(9903520314283042199192993792\, \sin^{93}(a) - 227780967228509970581438857216\, \sin^{91}(a) + 2534682230436816112855956848640\, \sin^{89}(a) - 18179149476905809336893639229440\, \sin^{87}(a) + 94455830823756434346276533829632\, \sin^{85}(a) - 378884624764955584962030590754816\, \sin^{83}(a) + 1220970054995401378433513352593408\, \sin^{81}(a) - 3247900638904023371202449312317440\, \sin^{79}(a) + 7272370543941348842060135415152640\, \sin^{77}(a) - 13907814517367573014292781846233088\, \sin^{75}(a) + 22972729336723223282537184299581440\, \sin^{73}(a) - 33062613744199688273005323296440320\, \sin^{71}(a) + 41748269895496049470740969711206400\, \sin^{69}(a) - 46505907490424374196751336629862400\, \sin^{67}(a) + 45903822080950130066525538075279360\, \sin^{65}(a) - 40286898703956232294672202192650240\, \sin^{63}(a) + 31522561365715753874797600513720320\, \sin^{61}(a) - 22034487127295580439602600817459200\, \sin^{59}(a) + 13779607995176439375789491958579200\, \sin^{57}(a) - 7716580477298806050442115496804352\, \sin^{55}(a) + 3871325002969164521927885646200832\, \sin^{53}(a) - 1739949724100296383306140385017856\, \sin^{51}(a) + 700263856763850343660141726924800\, \sin^{49}(a) - 252146446893535824906407431372800\, \sin^{47}(a) + 81122115801164353191615009914880\, \sin^{45}(a) - 23278520186421075263680828932096\, \sin^{43}(a) + 5944712592855834548389080465408\, \sin^{41}(a) - 1347336740227137736229697617920\, \sin^{39}(a) + 270123518535797906533064376320\, \sin^{37}(a) - 47719433247968542639528083456\, \sin^{35}(a) + 7394026766286792414197710848\, \sin^{33}(a) - 999499777931240756450689024\, \sin^{31}(a) + 117128880226317276146565120\, \sin^{29}(a) - 11811804612986789397790720\, \sin^{27}(a) + 1016162602734892911427584\, \sin^{25}(a) - 73813264121178661847040\, \sin^{23}(a) + 4471924287035009925120\, \sin^{21}(a) - 222642034205156966400\, \sin^{19}(a) + 8945438874314342400\, \sin^{17}(a) - 283585008602972160\, \sin^{15}(a) + 6892691181322240\, \sin^{13}(a) - 123706836664320\, \sin^{11}(a) + 1557663923200\, \sin^{9}(a) - 12785203200\, \sin^{7}(a) + 61020288\, \sin^{5}(a) - 138368\, \sin^{3}(a) + 94\, \sin(a)\big) \cdot \cos(a)$ $\sin(95a) = - 19807040628566084398385987584\, \sin^{95}(a) + 470417214928444504461667205120\, \sin^{93}(a) - 5409797971677111801309172858880\, \sin^{91}(a) + 40132468648582921786885983436800\, \sin^{89}(a) - 215877400038256485875611965849600\, \sin^{87}(a) + 897330392825686126289627071381504\, \sin^{85}(a) - 2999503279389231714282742176808960\, \sin^{83}(a) + 8285153944611652210798840606883840\, \sin^{81}(a) - 19284410043492638766514542791884800\, \sin^{79}(a) + 38381955648579341110872936913305600\, \sin^{77}(a) - 66062118957495971817890713769607168\, \sin^{75}(a) + 99200422135850282356410568566374400\, \sin^{73}(a) - 130872846070790432747312738048409600\, \sin^{71}(a) + 152541755387389411527707389329408000\, \sin^{69}(a) - 157787900413939841024692034994176000\, \sin^{67}(a) + 145362103256342078543997537238384640\, \sin^{65}(a) - 119601730527370064624808100259430400\, \sin^{63}(a) + 88077744992441077003110942611865600\, \sin^{61}(a) - 58146563252585559493395752157184000\, \sin^{59}(a) + 34449019987941098439473729896448000\, \sin^{57}(a) - 18326878633584664369800024304910336\, \sin^{55}(a) + 8756568459096919751979741342597120\, \sin^{53}(a) - 3756709631580185373047348558561280\, \sin^{51}(a) + 1446197095490560492341597044736000\, \sin^{49}(a) - 499039842810122986793931374592000\, \sin^{47}(a) + 154132020022212271064068518838272\, \sin^{45}(a) - 42528065725192349039416899010560\, \sin^{43}(a) + 10458290672616745964758567485440\, \sin^{41}(a) - 2285660541456751516818237030400\, \sin^{39}(a) + 442443694153462088286915788800\, \sin^{37}(a) - 75555769309283525845919465472\, \sin^{35}(a) + 11329557141891052892722298880\, \sin^{33}(a) - 1483632482866685497856491520\, \sin^{31}(a) + 168594600325759715665510400\, \sin^{29}(a) - 16501785856378602835148800\, \sin^{27}(a) + 1379077817997354665508864\, \sin^{25}(a) - 97392501270999623270400\, \sin^{23}(a) + 5740983882004404633600\, \sin^{21}(a) - 278302542756446208000\, \sin^{19}(a) + 10895085808459776000\, \sin^{17}(a) - 336757197716029440\, \sin^{15}(a) + 7985434905190400\, \sin^{13}(a) - 139906541465600\, \sin^{11}(a) + 1720675264000\, \sin^{9}(a) - 13802208000\, \sin^{7}(a) + 64410304\, \sin^{5}(a) - 142880\, \sin^{3}(a) + 95\, \sin(a)$ $\sin(96a) = \big( - 39614081257132168796771975168\, \sin^{95}(a) + 930930909542605966724141416448\, \sin^{93}(a) - 10591814976125713632036906860544\, \sin^{91}(a) + 77730255066729027460916010024960\, \sin^{89}(a) - 413575650599607162414330292469760\, \sin^{87}(a) + 1700204954827615818232977608933376\, \sin^{85}(a) - 5620121934013507843603453762863104\, \sin^{83}(a) + 15349337834227903043164167861174272\, \sin^{81}(a) - 35320919448081254161826636271452160\, \sin^{79}(a) + 69491540753217333379685738411458560\, \sin^{77}(a) - 118216423397624370621488645692981248\, \sin^{75}(a) + 175428114934977341430283952833167360\, \sin^{73}(a) - 228683078397381177221620152800378880\, \sin^{71}(a) + 263335240879282773584673808947609600\, \sin^{69}(a) - 269069893337455307852632733358489600\, \sin^{67}(a) + 244820384431734027021469536401489920\, \sin^{65}(a) - 198916562350783896954943998326210560\, \sin^{63}(a) + 144632928619166400131424284710010880\, \sin^{61}(a) - 94258639377875538547188903496908800\, \sin^{59}(a) + 55118431980705757503157967834316800\, \sin^{57}(a) - 28937176789870522689157933113016320\, \sin^{55}(a) + 13641811915224674982031597038993408\, \sin^{53}(a) - 5773469539060074362788556732104704\, \sin^{51}(a) + 2192130334217270641023052362547200\, \sin^{49}(a) - 745933238726710148681455317811200\, \sin^{47}(a) + 227141924243260188936522027761664\, \sin^{45}(a) - 61777611263963622815152969089024\, \sin^{43}(a) + 14971868752377657381128054505472\, \sin^{41}(a) - 3223984342686365297406776442880\, \sin^{39}(a) + 614763869771126270040767201280\, \sin^{37}(a) - 103392105370598509052310847488\, \sin^{35}(a) + 15265087517495313371246886912\, \sin^{33}(a) - 1967765187802130239262294016\, \sin^{31}(a) + 220060320425202155184455680\, \sin^{29}(a) - 21191767099770416272506880\, \sin^{27}(a) + 1741993033259816419590144\, \sin^{25}(a) - 120971738420820584693760\, \sin^{23}(a) + 7010043476973799342080\, \sin^{21}(a) - 333963051307735449600\, \sin^{19}(a) + 12844732742605209600\, \sin^{17}(a) - 389929386829086720\, \sin^{15}(a) + 9078178629058560\, \sin^{13}(a) - 156106246266880\, \sin^{11}(a) + 1883686604800\, \sin^{9}(a) - 14819212800\, \sin^{7}(a) + 67800320\, \sin^{5}(a) - 147392\, \sin^{3}(a) + 96\, \sin(a)\big) \cdot \cos(a)$ $\sin(97a) = 79228162514264337593543950336\, \sin^{97}(a) - 1921282940970910186643440795648\, \sin^{95}(a) + 22575074556408194693060429348864\, \sin^{93}(a) - 171234342114032370384596660912128\, \sin^{91}(a) + 942479342684089457963606621552640\, \sin^{89}(a) - 4011683810816189475419003836956672\, \sin^{87}(a) + 13743323384856561197383235672211456\, \sin^{85}(a) - 38939416257093590059252501071265792\, \sin^{83}(a) + 93055360620006662199182767658369024\, \sin^{81}(a) - 190340510359104536316510206573936640\, \sin^{79}(a) + 337033972653104066891475831295574016\, \sin^{77}(a) - 521226957707707452285654483282690048\, \sin^{75}(a) + 709021964528866754947397642700718080\, \sin^{73}(a) - 853163792482537468865275185447567360\, \sin^{71}(a) + 912268513046086751346905695282790400\, \sin^{69}(a) - 869992655124438828723512504525783040\, \sin^{67}(a) + 742111790308693769408829532217016320\, \sin^{65}(a) - 567497251412530529547928465813012480\, \sin^{63}(a) + 389705391001642800354115433801973760\, \sin^{61}(a) - 240607579464577032607297990505267200\, \sin^{59}(a) + 133662197553211461945158071998218240\, \sin^{57}(a) - 66831098776605730972579035999109120\, \sin^{55}(a) + 30073994449472578937660566199599104\, \sin^{53}(a) - 12174490114974504634575869630742528\, \sin^{51}(a) + 4429930050397401087067418315980800\, \sin^{49}(a) - 1447110483129817688442023316553728\, \sin^{47}(a) + 423707050992235352439281474863104\, \sin^{45}(a) - 110970894307490211353145148178432\, \sin^{43}(a) + 25933415517511299392311094411264\, \sin^{41}(a) - 5391835883458231618076850257920\, \sin^{39}(a) + 993868256129987469899240308736\, \sin^{37}(a) - 161758616466904119001196003328\, \sin^{35}(a) + 23136148268703834328296062976\, \sin^{33}(a) - 2892018533587979291037007872\, \sin^{31}(a) + 313909574724185427248414720\, \sin^{29}(a) - 29365734409681862549045248\, \sin^{27}(a) + 2346851725363919343058944\, \sin^{25}(a) - 158571062524589144801280\, \sin^{23}(a) + 8947029174558664949760\, \sin^{21}(a) - 415313025344235110400\, \sin^{19}(a) + 15574238450408816640\, \sin^{17}(a) - 461257933200261120\, \sin^{15}(a) + 10483134845460480\, \sin^{13}(a) - 176073324277760\, \sin^{11}(a) + 2076336371200\, \sin^{9}(a) - 15971818240\, \sin^{7}(a) + 71485120\, \sin^{5}(a) - 152096\, \sin^{3}(a) + 97\, \sin(a)$ $\sin(98a) = \big(158456325028528675187087900672\, \sin^{97}(a) - 3802951800684688204490109616128\, \sin^{95}(a) + 44219218203273783419396717281280\, \sin^{93}(a) - 331876869251939027137156414963712\, \sin^{91}(a) + 1807228430301449888466297233080320\, \sin^{89}(a) - 7609791971032771788423677381443584\, \sin^{87}(a) + 25786441814885506576533493735489536\, \sin^{85}(a) - 72258710580173672274901548379668480\, \sin^{83}(a) + 170761383405785421355201367455563776\, \sin^{81}(a) - 345360101270127818471193776876421120\, \sin^{79}(a) + 604576404552990800403265924179689472\, \sin^{77}(a) - 924237492017790533949820320872398848\, \sin^{75}(a) + 1242615814122756168464511332568268800\, \sin^{73}(a) - 1477644506567693760508930218094755840\, \sin^{71}(a) + 1561201785212890729109137581617971200\, \sin^{69}(a) - 1470915416911422349594392275693076480\, \sin^{67}(a) + 1239403196185653511796189528032542720\, \sin^{65}(a) - 936077940474277162140912933299814400\, \sin^{63}(a) + 634777853384119200576806582893936640\, \sin^{61}(a) - 386956519551278526667407077513625600\, \sin^{59}(a) + 212205963125717166387158176162119680\, \sin^{57}(a) - 104725020763340939256000138885201920\, \sin^{55}(a) + 46506176983720482893289535360204800\, \sin^{53}(a) - 18575510690888934906363182529380352\, \sin^{51}(a) + 6667729766577531533111784269414400\, \sin^{49}(a) - 2148287727532925228202591315296256\, \sin^{47}(a) + 620272177741210515942040921964544\, \sin^{45}(a) - 160164177351016799891137327267840\, \sin^{43}(a) + 36894962282644941403494134317056\, \sin^{41}(a) - 7559687424230097938746924072960\, \sin^{39}(a) + 1372972642488848669757713416192\, \sin^{37}(a) - 220125127563209728950081159168\, \sin^{35}(a) + 31007209019912355285345239040\, \sin^{33}(a) - 3816271879373828342811721728\, \sin^{31}(a) + 407758829023168699312373760\, \sin^{29}(a) - 37539701719593308825583616\, \sin^{27}(a) + 2951710417468022266527744\, \sin^{25}(a) - 196170386628357704908800\, \sin^{23}(a) + 10884014872143530557440\, \sin^{21}(a) - 496662999380734771200\, \sin^{19}(a) + 18303744158212423680\, \sin^{17}(a) - 532586479571435520\, \sin^{15}(a) + 11888091061862400\, \sin^{13}(a) - 196040402288640\, \sin^{11}(a) + 2268986137600\, \sin^{9}(a) - 17124423680\, \sin^{7}(a) + 75169920\, \sin^{5}(a) - 156800\, \sin^{3}(a) + 98\, \sin(a)\big) \cdot \cos(a)$ $\sin(99a) = - 316912650057057350374175801344\, \sin^{99}(a) + 7843588088912169421760851083264\, \sin^{97}(a) - 94123057066946033061130212999168\, \sin^{95}(a) + 729617100354017426420045835141120\, \sin^{93}(a) - 4106976256992745460822310635175936\, \sin^{91}(a) + 17891561459984353895816342607495168\, \sin^{89}(a) - 62780783761020367254495338396909568\, \sin^{87}(a) + 182346981405261796505486848558104576\, \sin^{85}(a) - 447100771714824597200953330599198720\, \sin^{83}(a) + 939187608731819817453607521005600768\, \sin^{81}(a) - 1709532501287132701432409195538284544\, \sin^{79}(a) + 2720593820488458601814696658808602624\, \sin^{77}(a) - 3812479654573385952543008823598645248\, \sin^{75}(a) + 4731498676852033102999485458625331200\, \sin^{73}(a) - 5224528791078631510370860413977886720\, \sin^{71}(a) + 5151965891202539406060154019339304960\, \sin^{69}(a) - 4550644571069712894057651102925455360\, \sin^{67}(a) + 3608850483011167578465375390447697920\, \sin^{65}(a) - 2574214336304262195887510566574489600\, \sin^{63}(a) + 1653763354869152654134311887013150720\, \sin^{61}(a) - 957717385889414353501832516846223360\, \sin^{59}(a) + 500199770224904749341158558096424960\, \sin^{57}(a) - 235631296717517113326000312491704320\, \sin^{55}(a) + 100089380899746256661644869579571200\, \sin^{53}(a) - 38311990799958428244374063966846976\, \sin^{51}(a) + 13202104937823512435561332853440512\, \sin^{49}(a) - 4090009327418453799847241157967872\, \sin^{47}(a) + 1137165659192219279227075023601664\, \sin^{45}(a) - 283147384959833271236117774991360\, \sin^{43}(a) + 62975883896238779292171022368768\, \sin^{41}(a) - 12473484249979661598932424720384\, \sin^{39}(a) + 2192327283974129327516348841984\, \sin^{37}(a) - 340506056699340049469656793088\, \sin^{35}(a) + 46510813529868532928017858560\, \sin^{33}(a) - 5556042883206014793211183104\, \sin^{31}(a) + 576687486761338589027500032\, \sin^{29}(a) - 51617089864440799635177472\, \sin^{27}(a) + 3948909882828840599814144\, \sin^{25}(a) - 255537740476413326131200\, \sin^{23}(a) + 13814326568489865707520\, \sin^{21}(a) - 614620461733659279360\, \sin^{19}(a) + 22098422825158901760\, \sin^{17}(a) - 627691208066334720\, \sin^{15}(a) + 13685128082841600\, \sin^{13}(a) - 220545452574720\, \sin^{11}(a) + 2495884751360\, \sin^{9}(a) - 18427368960\, \sin^{7}(a) + 79168320\, \sin^{5}(a) - 161700\, \sin^{3}(a) + 99\, \sin(a)$ $\sin(100a) = \big( - 633825300114114700748351602688\, \sin^{99}(a) + 15528719852795810168334614265856\, \sin^{97}(a) - 184443162333207377917770316382208\, \sin^{95}(a) + 1415014982504761069420694953000960\, \sin^{93}(a) - 7882075644733551894507464855388160\, \sin^{91}(a) + 33975894489667257903166387981910016\, \sin^{89}(a) - 117951775551007962720566999412375552\, \sin^{87}(a) + 338907520995638086434440203380719616\, \sin^{85}(a) - 821942832849475522127005112818728960\, \sin^{83}(a) + 1707613834057854213552013674555637760\, \sin^{81}(a) - 3073704901304137584393624614200147968\, \sin^{79}(a) + 4836611236423926403226127393437515776\, \sin^{77}(a) - 6700721817128981371136197326324891648\, \sin^{75}(a) + 8220381539581310037534459584682393600\, \sin^{73}(a) - 8971413075589569260232790609861017600\, \sin^{71}(a) + 8742729997192188083011170457060638720\, \sin^{69}(a) - 7630373725228003438520909930157834240\, \sin^{67}(a) + 5978297769836681645134561252862853120\, \sin^{65}(a) - 4212350732134247229634108199849164800\, \sin^{63}(a) + 2672748856354186107691817191132364800\, \sin^{61}(a) - 1528478252227550180336257956178821120\, \sin^{59}(a) + 788193577324092332295158940030730240\, \sin^{57}(a) - 366537572671693287396000486098206720\, \sin^{55}(a) + 153672584815772030430000203798937600\, \sin^{53}(a) - 58048470909027921582384945404313600\, \sin^{51}(a) + 19736480109069493338010881437466624\, \sin^{49}(a) - 6031730927303982371491891000639488\, \sin^{47}(a) + 1654059140643228042512109125238784\, \sin^{45}(a) - 406130592568649742581098222714880\, \sin^{43}(a) + 89056805509832617180847910420480\, \sin^{41}(a) - 17387281075729225259117925367808\, \sin^{39}(a) + 3011681925459409985274984267776\, \sin^{37}(a) - 460886985835470369989232427008\, \sin^{35}(a) + 62014418039824710570690478080\, \sin^{33}(a) - 7295813887038201243610644480\, \sin^{31}(a) + 745616144499508478742626304\, \sin^{29}(a) - 65694478009288290444771328\, \sin^{27}(a) + 4946109348189658933100544\, \sin^{25}(a) - 314905094324468947353600\, \sin^{23}(a) + 16744638264836200857600\, \sin^{21}(a) - 732577924086583787520\, \sin^{19}(a) + 25893101492105379840\, \sin^{17}(a) - 722795936561233920\, \sin^{15}(a) + 15482165103820800\, \sin^{13}(a) - 245050502860800\, \sin^{11}(a) + 2722783365120\, \sin^{9}(a) - 19730314240\, \sin^{7}(a) + 83166720\, \sin^{5}(a) - 166600\, \sin^{3}(a) + 100\, \sin(a)\big) \cdot \cos(a)$

## Decomposition of $\cos(na)$ as a sum of multiples of $\cos^{k}(a), \ k \le n,$ for $n = 0 \mathinner{\ldotp\ldotp} 100$

 $\cos(0) = 1$ $\cos(a) = \, \cos(a)$ $\cos(2a) = 2\, \cos^{2}(a) - 1$ $\cos(3a) = 4\, \cos^{3}(a) - 3\, \cos(a)$ $\cos(4a) = 8\, \cos^{4}(a) - 8\, \cos^{2}(a) + 1$ $\cos(5a) = 16\, \cos^{5}(a) - 20\, \cos^{3}(a) + 5\, \cos(a)$ $\cos(6a) = 32\, \cos^{6}(a) - 48\, \cos^{4}(a) + 18\, \cos^{2}(a) - 1$ $\cos(7a) = 64\, \cos^{7}(a) - 112\, \cos^{5}(a) + 56\, \cos^{3}(a) - 7\, \cos(a)$ $\cos(8a) = 128\, \cos^{8}(a) - 256\, \cos^{6}(a) + 160\, \cos^{4}(a) - 32\, \cos^{2}(a) + 1$ $\cos(9a) = 256\, \cos^{9}(a) - 576\, \cos^{7}(a) + 432\, \cos^{5}(a) - 120\, \cos^{3}(a) + 9\, \cos(a)$ $\cos(10a) = 512\, \cos^{10}(a) - 1280\, \cos^{8}(a) + 1120\, \cos^{6}(a) - 400\, \cos^{4}(a) + 50\, \cos^{2}(a) - 1$ $\cos(11a) = 1024\, \cos^{11}(a) - 2816\, \cos^{9}(a) + 2816\, \cos^{7}(a) - 1232\, \cos^{5}(a) + 220\, \cos^{3}(a) - 11\, \cos(a)$ $\cos(12a) = 2048\, \cos^{12}(a) - 6144\, \cos^{10}(a) + 6912\, \cos^{8}(a) - 3584\, \cos^{6}(a) + 840\, \cos^{4}(a) - 72\, \cos^{2}(a) + 1$ $\cos(13a) = 4096\, \cos^{13}(a) - 13312\, \cos^{11}(a) + 16640\, \cos^{9}(a) - 9984\, \cos^{7}(a) + 2912\, \cos^{5}(a) - 364\, \cos^{3}(a) + 13\, \cos(a)$ $\cos(14a) = 8192\, \cos^{14}(a) - 28672\, \cos^{12}(a) + 39424\, \cos^{10}(a) - 26880\, \cos^{8}(a) + 9408\, \cos^{6}(a) - 1568\, \cos^{4}(a) + 98\, \cos^{2}(a) - 1$ $\cos(15a) = 16384\, \cos^{15}(a) - 61440\, \cos^{13}(a) + 92160\, \cos^{11}(a) - 70400\, \cos^{9}(a) + 28800\, \cos^{7}(a) - 6048\, \cos^{5}(a) + 560\, \cos^{3}(a) - 15\, \cos(a)$ $\cos(16a) = 32768\, \cos^{16}(a) - 131072\, \cos^{14}(a) + 212992\, \cos^{12}(a) - 180224\, \cos^{10}(a) + 84480\, \cos^{8}(a) - 21504\, \cos^{6}(a) + 2688\, \cos^{4}(a) - 128\, \cos^{2}(a) + 1$ $\cos(17a) = 65536\, \cos^{17}(a) - 278528\, \cos^{15}(a) + 487424\, \cos^{13}(a) - 452608\, \cos^{11}(a) + 239360\, \cos^{9}(a) - 71808\, \cos^{7}(a) + 11424\, \cos^{5}(a) - 816\, \cos^{3}(a) + 17\, \cos(a)$ $\cos(18a) = 131072\, \cos^{18}(a) - 589824\, \cos^{16}(a) + 1105920\, \cos^{14}(a) - 1118208\, \cos^{12}(a) + 658944\, \cos^{10}(a) - 228096\, \cos^{8}(a) + 44352\, \cos^{6}(a) - 4320\, \cos^{4}(a) + 162\, \cos^{2}(a) - 1$ $\cos(19a) = 262144\, \cos^{19}(a) - 1245184\, \cos^{17}(a) + 2490368\, \cos^{15}(a) - 2723840\, \cos^{13}(a) + 1770496\, \cos^{11}(a) - 695552\, \cos^{9}(a) + 160512\, \cos^{7}(a) - 20064\, \cos^{5}(a) + 1140\, \cos^{3}(a) - 19\, \cos(a)$ $\cos(20a) = 524288\, \cos^{20}(a) - 2621440\, \cos^{18}(a) + 5570560\, \cos^{16}(a) - 6553600\, \cos^{14}(a) + 4659200\, \cos^{12}(a) - 2050048\, \cos^{10}(a) + 549120\, \cos^{8}(a) - 84480\, \cos^{6}(a) + 6600\, \cos^{4}(a) - 200\, \cos^{2}(a) + 1$ $\cos(21a) = 1048576\, \cos^{21}(a) - 5505024\, \cos^{19}(a) + 12386304\, \cos^{17}(a) - 15597568\, \cos^{15}(a) + 12042240\, \cos^{13}(a) - 5870592\, \cos^{11}(a) + 1793792\, \cos^{9}(a) - 329472\, \cos^{7}(a) + 33264\, \cos^{5}(a) - 1540\, \cos^{3}(a) + 21\, \cos(a)$ $\cos(22a) = 2097152\, \cos^{22}(a) - 11534336\, \cos^{20}(a) + 27394048\, \cos^{18}(a) - 36765696\, \cos^{16}(a) + 30638080\, \cos^{14}(a) - 16400384\, \cos^{12}(a) + 5637632\, \cos^{10}(a) - 1208064\, \cos^{8}(a) + 151008\, \cos^{6}(a) - 9680\, \cos^{4}(a) + 242\, \cos^{2}(a) - 1$ $\cos(23a) = 4194304\, \cos^{23}(a) - 24117248\, \cos^{21}(a) + 60293120\, \cos^{19}(a) - 85917696\, \cos^{17}(a) + 76873728\, \cos^{15}(a) - 44843008\, \cos^{13}(a) + 17145856\, \cos^{11}(a) - 4209920\, \cos^{9}(a) + 631488\, \cos^{7}(a) - 52624\, \cos^{5}(a) + 2024\, \cos^{3}(a) - 23\, \cos(a)$ $\cos(24a) = 8388608\, \cos^{24}(a) - 50331648\, \cos^{22}(a) + 132120576\, \cos^{20}(a) - 199229440\, \cos^{18}(a) + 190513152\, \cos^{16}(a) - 120324096\, \cos^{14}(a) + 50692096\, \cos^{12}(a) - 14057472\, \cos^{10}(a) + 2471040\, \cos^{8}(a) - 256256\, \cos^{6}(a) + 13728\, \cos^{4}(a) - 288\, \cos^{2}(a) + 1$ $\cos(25a) = 16777216\, \cos^{25}(a) - 104857600\, \cos^{23}(a) + 288358400\, \cos^{21}(a) - 458752000\, \cos^{19}(a) + 466944000\, \cos^{17}(a) - 317521920\, \cos^{15}(a) + 146227200\, \cos^{13}(a) - 45260800\, \cos^{11}(a) + 9152000\, \cos^{9}(a) - 1144000\, \cos^{7}(a) + 80080\, \cos^{5}(a) - 2600\, \cos^{3}(a) + 25\, \cos(a)$ $\cos(26a) = 33554432\, \cos^{26}(a) - 218103808\, \cos^{24}(a) + 627048448\, \cos^{22}(a) - 1049624576\, \cos^{20}(a) + 1133117440\, \cos^{18}(a) - 825556992\, \cos^{16}(a) + 412778496\, \cos^{14}(a) - 141213696\, \cos^{12}(a) + 32361472\, \cos^{10}(a) - 4759040\, \cos^{8}(a) + 416416\, \cos^{6}(a) - 18928\, \cos^{4}(a) + 338\, \cos^{2}(a) - 1$ $\cos(27a) = 67108864\, \cos^{27}(a) - 452984832\, \cos^{25}(a) + 1358954496\, \cos^{23}(a) - 2387607552\, \cos^{21}(a) + 2724986880\, \cos^{19}(a) - 2118057984\, \cos^{17}(a) + 1143078912\, \cos^{15}(a) - 428654592\, \cos^{13}(a) + 109983744\, \cos^{11}(a) - 18670080\, \cos^{9}(a) + 1976832\, \cos^{7}(a) - 117936\, \cos^{5}(a) + 3276\, \cos^{3}(a) - 27\, \cos(a)$ $\cos(28a) = 134217728\, \cos^{28}(a) - 939524096\, \cos^{26}(a) + 2936012800\, \cos^{24}(a) - 5402263552\, \cos^{22}(a) + 6499598336\, \cos^{20}(a) - 5369233408\, \cos^{18}(a) + 3111714816\, \cos^{16}(a) - 1270087680\, \cos^{14}(a) + 361181184\, \cos^{12}(a) - 69701632\, \cos^{10}(a) + 8712704\, \cos^{8}(a) - 652288\, \cos^{6}(a) + 25480\, \cos^{4}(a) - 392\, \cos^{2}(a) + 1$ $\cos(29a) = 268435456\, \cos^{29}(a) - 1946157056\, \cos^{27}(a) + 6325010432\, \cos^{25}(a) - 12163481600\, \cos^{23}(a) + 15386804224\, \cos^{21}(a) - 13463453696\, \cos^{19}(a) + 8341487616\, \cos^{17}(a) - 3683254272\, \cos^{15}(a) + 1151016960\, \cos^{13}(a) - 249387008\, \cos^{11}(a) + 36095488\, \cos^{9}(a) - 3281408\, \cos^{7}(a) + 168896\, \cos^{5}(a) - 4060\, \cos^{3}(a) + 29\, \cos(a)$ $\cos(30a) = 536870912\, \cos^{30}(a) - 4026531840\, \cos^{28}(a) + 13589544960\, \cos^{26}(a) - 27262976000\, \cos^{24}(a) + 36175872000\, \cos^{22}(a) - 33426505728\, \cos^{20}(a) + 22052208640\, \cos^{18}(a) - 10478223360\, \cos^{16}(a) + 3572121600\, \cos^{14}(a) - 859955200\, \cos^{12}(a) + 141892608\, \cos^{10}(a) - 15275520\, \cos^{8}(a) + 990080\, \cos^{6}(a) - 33600\, \cos^{4}(a) + 450\, \cos^{2}(a) - 1$ $\cos(31a) = 1073741824\, \cos^{31}(a) - 8321499136\, \cos^{29}(a) + 29125246976\, \cos^{27}(a) - 60850962432\, \cos^{25}(a) + 84515225600\, \cos^{23}(a) - 82239815680\, \cos^{21}(a) + 57567870976\, \cos^{19}(a) - 29297934336\, \cos^{17}(a) + 10827497472\, \cos^{15}(a) - 2870927360\, \cos^{13}(a) + 533172224\, \cos^{11}(a) - 66646528\, \cos^{9}(a) + 5261568\, \cos^{7}(a) - 236096\, \cos^{5}(a) + 4960\, \cos^{3}(a) - 31\, \cos(a)$ $\cos(32a) = 2147483648\, \cos^{32}(a) - 17179869184\, \cos^{30}(a) + 62277025792\, \cos^{28}(a) - 135291469824\, \cos^{26}(a) + 196293427200\, \cos^{24}(a) - 200655503360\, \cos^{22}(a) + 148562247680\, \cos^{20}(a) - 80648077312\, \cos^{18}(a) + 32133218304\, \cos^{16}(a) - 9313976320\, \cos^{14}(a) + 1926299648\, \cos^{12}(a) - 275185664\, \cos^{10}(a) + 25798656\, \cos^{8}(a) - 1462272\, \cos^{6}(a) + 43520\, \cos^{4}(a) - 512\, \cos^{2}(a) + 1$ $\cos(33a) = 4294967296\, \cos^{33}(a) - 35433480192\, \cos^{31}(a) + 132875550720\, \cos^{29}(a) - 299708186624\, \cos^{27}(a) + 453437816832\, \cos^{25}(a) - 485826232320\, \cos^{23}(a) + 379364311040\, \cos^{21}(a) - 218864025600\, \cos^{19}(a) + 93564370944\, \cos^{17}(a) - 29455450112\, \cos^{15}(a) + 6723526656\, \cos^{13}(a) - 1083543552\, \cos^{11}(a) + 118243840\, \cos^{9}(a) - 8186112\, \cos^{7}(a) + 323136\, \cos^{5}(a) - 5984\, \cos^{3}(a) + 33\, \cos(a)$ $\cos(34a) = 8589934592\, \cos^{34}(a) - 73014444032\, \cos^{32}(a) + 282930970624\, \cos^{30}(a) - 661693399040\, \cos^{28}(a) + 1042167103488\, \cos^{26}(a) - 1167945891840\, \cos^{24}(a) + 959384125440\, \cos^{22}(a) - 586290298880\, \cos^{20}(a) + 267776819200\, \cos^{18}(a) - 91044118528\, \cos^{16}(a) + 22761029632\, \cos^{14}(a) - 4093386752\, \cos^{12}(a) + 511673344\, \cos^{10}(a) - 42170880\, \cos^{8}(a) + 2108544\, \cos^{6}(a) - 55488\, \cos^{4}(a) + 578\, \cos^{2}(a) - 1$ $\cos(35a) = 17179869184\, \cos^{35}(a) - 150323855360\, \cos^{33}(a) + 601295421440\, \cos^{31}(a) - 1456262348800\, \cos^{29}(a) + 2384042393600\, \cos^{27}(a) - 2789329600512\, \cos^{25}(a) + 2404594483200\, \cos^{23}(a) - 1551944908800\, \cos^{21}(a) + 754417664000\, \cos^{19}(a) - 275652608000\, \cos^{17}(a) + 74977509376\, \cos^{15}(a) - 14910300160\, \cos^{13}(a) + 2106890240\, \cos^{11}(a) - 202585600\, \cos^{9}(a) + 12403200\, \cos^{7}(a) - 434112\, \cos^{5}(a) + 7140\, \cos^{3}(a) - 35\, \cos(a)$ $\cos(36a) = 34359738368\, \cos^{36}(a) - 309237645312\, \cos^{34}(a) + 1275605286912\, \cos^{32}(a) - 3195455668224\, \cos^{30}(a) + 5429778186240\, \cos^{28}(a) - 6620826304512\, \cos^{26}(a) + 5977134858240\, \cos^{24}(a) - 4063273943040\, \cos^{22}(a) + 2095125626880\, \cos^{20}(a) - 819082035200\, \cos^{18}(a) + 240999137280\, \cos^{16}(a) - 52581629952\, \cos^{14}(a) + 8307167232\, \cos^{12}(a) - 916844544\, \cos^{10}(a) + 66977280\, \cos^{8}(a) - 2976768\, \cos^{6}(a) + 69768\, \cos^{4}(a) - 648\, \cos^{2}(a) + 1$ $\cos(37a) = 68719476736\, \cos^{37}(a) - 635655159808\, \cos^{35}(a) + 2701534429184\, \cos^{33}(a) - 6992206757888\, \cos^{31}(a) + 12315818721280\, \cos^{29}(a) - 15625695002624\, \cos^{27}(a) + 14743599316992\, \cos^{25}(a) - 10531142369280\, \cos^{23}(a) + 5742196162560\, \cos^{21}(a) - 2392581734400\, \cos^{19}(a) + 757650882560\, \cos^{17}(a) - 180140769280\, \cos^{15}(a) + 31524634624\, \cos^{13}(a) - 3940579328\, \cos^{11}(a) + 336540160\, \cos^{9}(a) - 18356736\, \cos^{7}(a) + 573648\, \cos^{5}(a) - 8436\, \cos^{3}(a) + 37\, \cos(a)$ $\cos(38a) = 137438953472\, \cos^{38}(a) - 1305670057984\, \cos^{36}(a) + 5712306503680\, \cos^{34}(a) - 15260018802688\, \cos^{32}(a) + 27827093110784\, \cos^{30}(a) - 36681168191488\, \cos^{28}(a) + 36108024938496\, \cos^{26}(a) - 27039419596800\, \cos^{24}(a) + 15547666268160\, \cos^{22}(a) - 6880289095680\, \cos^{20}(a) + 2334383800320\, \cos^{18}(a) - 601280675840\, \cos^{16}(a) + 115630899200\, \cos^{14}(a) - 16188325888\, \cos^{12}(a) + 1589924864\, \cos^{10}(a) - 103690752\, \cos^{8}(a) + 4124064\, \cos^{6}(a) - 86640\, \cos^{4}(a) + 722\, \cos^{2}(a) - 1$ $\cos(39a) = 274877906944\, \cos^{39}(a) - 2680059592704\, \cos^{37}(a) + 12060268167168\, \cos^{35}(a) - 33221572034560\, \cos^{33}(a) + 62646392979456\, \cos^{31}(a) - 85678155104256\, \cos^{29}(a) + 87841744879616\, \cos^{27}(a) - 68822438510592\, \cos^{25}(a) + 41626474905600\, \cos^{23}(a) - 19502774353920\, \cos^{21}(a) + 7061349335040\, \cos^{19}(a) - 1960212234240\, \cos^{17}(a) + 411402567680\, \cos^{15}(a) - 63901286400\, \cos^{13}(a) + 7120429056\, \cos^{11}(a) - 543921664\, \cos^{9}(a) + 26604864\, \cos^{7}(a) - 746928\, \cos^{5}(a) + 9880\, \cos^{3}(a) - 39\, \cos(a)$ $\cos(40a) = 549755813888\, \cos^{40}(a) - 5497558138880\, \cos^{38}(a) + 25426206392320\, \cos^{36}(a) - 72155450572800\, \cos^{34}(a) + 140552804761600\, \cos^{32}(a) - 199183403319296\, \cos^{30}(a) + 212364657950720\, \cos^{28}(a) - 173752901959680\, \cos^{26}(a) + 110292369408000\, \cos^{24}(a) - 54553214976000\, \cos^{22}(a) + 21002987765760\, \cos^{20}(a) - 6254808268800\, \cos^{18}(a) + 1424085811200\, \cos^{16}(a) - 243433472000\, \cos^{14}(a) + 30429184000\, \cos^{12}(a) - 2677768192\, \cos^{10}(a) + 156900480\, \cos^{8}(a) - 5617920\, \cos^{6}(a) + 106400\, \cos^{4}(a) - 800\, \cos^{2}(a) + 1$ $\cos(41a) = 1099511627776\, \cos^{41}(a) - 11269994184704\, \cos^{39}(a) + 53532472377344\, \cos^{37}(a) - 156371169312768\, \cos^{35}(a) + 314327181557760\, \cos^{33}(a) - 461013199618048\, \cos^{31}(a) + 510407471005696\, \cos^{29}(a) - 435347548798976\, \cos^{27}(a) + 289407177326592\, \cos^{25}(a) - 150732904857600\, \cos^{23}(a) + 61508749885440\, \cos^{21}(a) - 19570965872640\, \cos^{19}(a) + 4808383856640\, \cos^{17}(a) - 898269511680\, \cos^{15}(a) + 124759654400\, \cos^{13}(a) - 12475965440\, \cos^{11}(a) + 857722624\, \cos^{9}(a) - 37840704\, \cos^{7}(a) + 959728\, \cos^{5}(a) - 11480\, \cos^{3}(a) + 41\, \cos(a)$ $\cos(42a) = 2199023255552\, \cos^{42}(a) - 23089744183296\, \cos^{40}(a) + 112562502893568\, \cos^{38}(a) - 338168545017856\, \cos^{36}(a) + 700809813688320\, \cos^{34}(a) - 1062579203997696\, \cos^{32}(a) + 1219998345330688\, \cos^{30}(a) - 1083059755548672\, \cos^{28}(a) + 752567256612864\, \cos^{26}(a) - 411758179123200\, \cos^{24}(a) + 177570714746880\, \cos^{22}(a) - 60144919511040\, \cos^{20}(a) + 15871575982080\, \cos^{18}(a) - 3220624834560\, \cos^{16}(a) + 492952780800\, \cos^{14}(a) - 55381114880\, \cos^{12}(a) + 4393213440\, \cos^{10}(a) - 232581888\, \cos^{8}(a) + 7537376\, \cos^{6}(a) - 129360\, \cos^{4}(a) + 882\, \cos^{2}(a) - 1$ $\cos(43a) = 4398046511104\, \cos^{43}(a) - 47278999994368\, \cos^{41}(a) + 236394999971840\, \cos^{39}(a) - 729869562413056\, \cos^{37}(a) + 1557990796689408\, \cos^{35}(a) - 2439485589553152\, \cos^{33}(a) + 2901009890279424\, \cos^{31}(a) - 2676526982103040\, \cos^{29}(a) + 1940482062024704\, \cos^{27}(a) - 1112923535572992\, \cos^{25}(a) + 505874334351360\, \cos^{23}(a) - 181798588907520\, \cos^{21}(a) + 51314117836800\, \cos^{19}(a) - 11249633525760\, \cos^{17}(a) + 1884175073280\, \cos^{15}(a) - 235521884160\, \cos^{13}(a) + 21262392320\, \cos^{11}(a) - 1322886400\, \cos^{9}(a) + 52915456\, \cos^{7}(a) - 1218448\, \cos^{5}(a) + 13244\, \cos^{3}(a) - 43\, \cos(a)$ $\cos(44a) = 8796093022208\, \cos^{44}(a) - 96757023244288\, \cos^{42}(a) + 495879744126976\, \cos^{40}(a) - 1572301627719680\, \cos^{38}(a) + 3454150138396672\, \cos^{36}(a) - 5579780992794624\, \cos^{34}(a) + 6864598984556544\, \cos^{32}(a) - 6573052309536768\, \cos^{30}(a) + 4964023879598080\, \cos^{28}(a) - 2978414327758848\, \cos^{26}(a) + 1423506847825920\, \cos^{24}(a) - 541167892561920\, \cos^{22}(a) + 162773155184640\, \cos^{20}(a) - 38370843033600\, \cos^{18}(a) + 6988974981120\, \cos^{16}(a) - 963996549120\, \cos^{14}(a) + 97905899520\, \cos^{12}(a) - 7038986240\, \cos^{10}(a) + 338412800\, \cos^{8}(a) - 9974272\, \cos^{6}(a) + 155848\, \cos^{4}(a) - 968\, \cos^{2}(a) + 1$ $\cos(45a) = 17592186044416\, \cos^{45}(a) - 197912092999680\, \cos^{43}(a) + 1039038488248320\, \cos^{41}(a) - 3380998255411200\, \cos^{39}(a) + 7638169839206400\, \cos^{37}(a) - 12717552782278656\, \cos^{35}(a) + 16168683558666240\, \cos^{33}(a) - 16047114509352960\, \cos^{31}(a) + 12604574741299200\, \cos^{29}(a) - 7897310717542400\, \cos^{27}(a) + 3959937231224832\, \cos^{25}(a) - 1588210119475200\, \cos^{23}(a) + 507344899276800\, \cos^{21}(a) - 128055803904000\, \cos^{19}(a) + 25227583488000\, \cos^{17}(a) - 3812168171520\, \cos^{15}(a) + 431333683200\, \cos^{13}(a) - 35340364800\, \cos^{11}(a) + 1999712000\, \cos^{9}(a) - 72864000\, \cos^{7}(a) + 1530144\, \cos^{5}(a) - 15180\, \cos^{3}(a) + 45\, \cos(a)$ $\cos(46a) = 35184372088832\, \cos^{46}(a) - 404620279021568\, \cos^{44}(a) + 2174833999740928\, \cos^{42}(a) - 7257876254949376\, \cos^{40}(a) + 16848641306132480\, \cos^{38}(a) - 28889255702953984\, \cos^{36}(a) + 37917148110127104\, \cos^{34}(a) - 38958828003262464\, \cos^{32}(a) + 31782201792135168\, \cos^{30}(a) - 20758645314682880\, \cos^{28}(a) + 10898288790208512\, \cos^{26}(a) - 4599927086776320\, \cos^{24}(a) + 1555857691115520\, \cos^{22}(a) - 418884762992640\, \cos^{20}(a) + 88826010009600\, \cos^{18}(a) - 14613311324160\, \cos^{16}(a) + 1826663915520\, \cos^{14}(a) - 168586629120\, \cos^{12}(a) + 11038410240\, \cos^{10}(a) - 484140800\, \cos^{8}(a) + 13034560\, \cos^{6}(a) - 186208\, \cos^{4}(a) + 1058\, \cos^{2}(a) - 1$ $\cos(47a) = 70368744177664\, \cos^{47}(a) - 826832744087552\, \cos^{45}(a) + 4547580092481536\, \cos^{43}(a) - 15554790998147072\, \cos^{41}(a) + 37078280867676160\, \cos^{39}(a) - 65416681245114368\, \cos^{37}(a) + 88551849002532864\, \cos^{35}(a) - 94086339565191168\, \cos^{33}(a) + 79611518093623296\, \cos^{31}(a) - 54121865370664960\, \cos^{29}(a) + 29693888297959424\, \cos^{27}(a) - 13159791404777472\, \cos^{25}(a) + 4699925501706240\, \cos^{23}(a) - 1345114425262080\, \cos^{21}(a) + 305707823923200\, \cos^{19}(a) - 54454206136320\, \cos^{17}(a) + 7465496002560\, \cos^{15}(a) - 768506941440\, \cos^{13}(a) + 57417185280\, \cos^{11}(a) - 2967993600\, \cos^{9}(a) + 98933120\, \cos^{7}(a) - 1902560\, \cos^{5}(a) + 17296\, \cos^{3}(a) - 47\, \cos(a)$ $\cos(48a) = 140737488355328\, \cos^{48}(a) - 1688849860263936\, \cos^{46}(a) + 9499780463984640\, \cos^{44}(a) - 33284415996035072\, \cos^{42}(a) + 81414437990301696\, \cos^{40}(a) - 147682003796361216\, \cos^{38}(a) + 205992953708019712\, \cos^{36}(a) - 226089827240509440\, \cos^{34}(a) + 198181864190509056\, \cos^{32}(a) - 140025932533465088\, \cos^{30}(a) + 80146421910601728\, \cos^{28}(a) - 37217871599763456\, \cos^{26}(a) + 13999778090188800\, \cos^{24}(a) - 4246086541639680\, \cos^{22}(a) + 1030300410839040\, \cos^{20}(a) - 197734422282240\, \cos^{18}(a) + 29544303329280\, \cos^{16}(a) - 3363677798400\, \cos^{14}(a) + 283420999680\, \cos^{12}(a) - 16974397440\, \cos^{10}(a) + 682007040\, \cos^{8}(a) - 16839680\, \cos^{6}(a) + 220800\, \cos^{4}(a) - 1152\, \cos^{2}(a) + 1$ $\cos(49a) = 281474976710656\, \cos^{49}(a) - 3448068464705536\, \cos^{47}(a) + 19826393672056832\, \cos^{45}(a) - 71116412084551680\, \cos^{43}(a) + 178383666978750464\, \cos^{41}(a) - 332442288460398592\, \cos^{39}(a) + 477402588661153792\, \cos^{37}(a) - 540731503483551744\, \cos^{35}(a) + 490450067946209280\, \cos^{33}(a) - 359663383160553472\, \cos^{31}(a) + 214414709191868416\, \cos^{29}(a) - 104129631497486336\, \cos^{27}(a) + 41159347585155072\, \cos^{25}(a) - 13192098584985600\, \cos^{23}(a) + 3405715246940160\, \cos^{21}(a) - 701176668487680\, \cos^{19}(a) + 113542812794880\, \cos^{17}(a) - 14192851599360\, \cos^{15}(a) + 1335348940800\, \cos^{13}(a) - 91365980160\, \cos^{11}(a) + 4332007680\, \cos^{9}(a) - 132612480\, \cos^{7}(a) + 2344160\, \cos^{5}(a) - 19600\, \cos^{3}(a) + 49\, \cos(a)$ $\cos(50a) = 562949953421312\, \cos^{50}(a) - 7036874417766400\, \cos^{48}(a) + 41341637204377600\, \cos^{46}(a) - 151732604633088000\, \cos^{44}(a) + 390051749953536000\, \cos^{42}(a) - 746299014911098880\, \cos^{40}(a) + 1102487181118668800\, \cos^{38}(a) - 1287455960675123200\, \cos^{36}(a) + 1206989963132928000\, \cos^{34}(a) - 917508630511616000\, \cos^{32}(a) + 568855350917201920\, \cos^{30}(a) - 288405684905574400\, \cos^{28}(a) + 119536566770073600\, \cos^{26}(a) - 40383975260160000\, \cos^{24}(a) + 11057517035520000\, \cos^{22}(a) - 2432653747814400\, \cos^{20}(a) + 424820047872000\, \cos^{18}(a) - 57930006528000\, \cos^{16}(a) + 6034375680000\, \cos^{14}(a) - 466152960000\, \cos^{12}(a) + 25638412800\, \cos^{10}(a) - 947232000\, \cos^{8}(a) + 21528000\, \cos^{6}(a) - 260000\, \cos^{4}(a) + 1250\, \cos^{2}(a) - 1$ $\cos(51a) = 1125899906842624\, \cos^{51}(a) - 14355223812243456\, \cos^{49}(a) + 86131342873460736\, \cos^{47}(a) - 323291602938232832\, \cos^{45}(a) + 851219911991623680\, \cos^{43}(a) - 1670981696800948224\, \cos^{41}(a) + 2537416650697736192\, \cos^{39}(a) - 3052314510011400192\, \cos^{37}(a) + 2954711429749407744\, \cos^{35}(a) - 2325467328969441280\, \cos^{33}(a) + 1497374084994957312\, \cos^{31}(a) - 791226079003017216\, \cos^{29}(a) + 343202765037633536\, \cos^{27}(a) - 121927298105475072\, \cos^{25}(a) + 35307132656025600\, \cos^{23}(a) - 8271022742568960\, \cos^{21}(a) + 1550816764231680\, \cos^{19}(a) - 229402825850880\, \cos^{17}(a) + 26261602959360\, \cos^{15}(a) - 2267654860800\, \cos^{13}(a) + 142642805760\, \cos^{11}(a) - 6226471680\, \cos^{9}(a) + 175668480\, \cos^{7}(a) - 2864160\, \cos^{5}(a) + 22100\, \cos^{3}(a) - 51\, \cos(a)$ $\cos(52a) = 2251799813685248\, \cos^{52}(a) - 29273397577908224\, \cos^{50}(a) + 179299560164687872\, \cos^{48}(a) - 687924843080843264\, \cos^{46}(a) + 1854172428616335360\, \cos^{44}(a) - 3732015143555432448\, \cos^{42}(a) + 5821132316306571264\, \cos^{40}(a) - 7207116201141469184\, \cos^{38}(a) + 7196878820173938688\, \cos^{36}(a) - 5857924621071810560\, \cos^{34}(a) + 3912256800501530624\, \cos^{32}(a) - 2151307508923236352\, \cos^{30}(a) + 974811214980841472\, \cos^{28}(a) - 363391162981023744\, \cos^{26}(a) + 110998240572211200\, \cos^{24}(a) - 27599562520657920\, \cos^{22}(a) + 5534287276277760\, \cos^{20}(a) - 883625699573760\, \cos^{18}(a) + 110453212446720\, \cos^{16}(a) - 10569685401600\, \cos^{14}(a) + 751438571520\, \cos^{12}(a) - 38091356160\, \cos^{10}(a) + 1298568960\, \cos^{8}(a) - 27256320\, \cos^{6}(a) + 304200\, \cos^{4}(a) - 1352\, \cos^{2}(a) + 1$ $\cos(53a) = 4503599627370496\, \cos^{53}(a) - 59672695062659072\, \cos^{51}(a) + 372954344141619200\, \cos^{49}(a) - 1461981029035147264\, \cos^{47}(a) + 4031636460170903552\, \cos^{45}(a) - 8315250199102488576\, \cos^{43}(a) + 13313246329414090752\, \cos^{41}(a) - 16951649052980674560\, \cos^{39}(a) + 17446072150359277568\, \cos^{37}(a) - 14670560671893028864\, \cos^{35}(a) + 10149980929972502528\, \cos^{33}(a) - 5799989102841430016\, \cos^{31}(a) + 2740848508964700160\, \cos^{29}(a) - 1069985090999681024\, \cos^{27}(a) + 343923779249897472\, \cos^{25}(a) - 90506257697341440\, \cos^{23}(a) + 19339597295124480\, \cos^{21}(a) - 3318068163379200\, \cos^{19}(a) + 450309250744320\, \cos^{17}(a) - 47400973762560\, \cos^{15}(a) + 3770532003840\, \cos^{13}(a) - 218825518080\, \cos^{11}(a) + 8823609600\, \cos^{9}(a) - 230181120\, \cos^{7}(a) + 3472560\, \cos^{5}(a) - 24804\, \cos^{3}(a) + 53\, \cos(a)$ $\cos(54a) = 9007199254740992\, \cos^{54}(a) - 121597189939003392\, \cos^{52}(a) + 775182085861146624\, \cos^{50}(a) - 3103261618234982400\, \cos^{48}(a) + 8751197763422650368\, \cos^{46}(a) - 18484672826821312512\, \cos^{44}(a) + 30358507802383613952\, \cos^{42}(a) - 39724430422267920384\, \cos^{40}(a) + 42099260501860024320\, \cos^{38}(a) - 36538000163959996416\, \cos^{36}(a) + 26157886481016815616\, \cos^{34}(a) - 15512235006184390656\, \cos^{32}(a) + 7633004526852636672\, \cos^{30}(a) - 3114781396980203520\, \cos^{28}(a) + 1051238721480818688\, \cos^{26}(a) - 292010755966894080\, \cos^{24}(a) + 66278757110906880\, \cos^{22}(a) - 12170423603036160\, \cos^{20}(a) + 1784244201062400\, \cos^{18}(a) - 205255159971840\, \cos^{16}(a) + 18110749409280\, \cos^{14}(a) - 1189089607680\, \cos^{12}(a) + 55738575360\, \cos^{10}(a) - 1758931200\, \cos^{8}(a) + 34201440\, \cos^{6}(a) - 353808\, \cos^{4}(a) + 1458\, \cos^{2}(a) - 1$ $\cos(55a) = 18014398509481984\, \cos^{55}(a) - 247697979505377280\, \cos^{53}(a) + 1610036866784952320\, \cos^{51}(a) - 6579477580611584000\, \cos^{49}(a) + 18964376555880448000\, \cos^{47}(a) - 41000982113813528576\, \cos^{45}(a) + 69032265803869716480\, \cos^{43}(a) - 92762107173949931520\, \cos^{41}(a) + 101150170056700723200\, \cos^{39}(a) - 90522072478279270400\, \cos^{37}(a) + 66986333633926660096\, \cos^{35}(a) - 41174450942341283840\, \cos^{33}(a) + 21065998156546703360\, \cos^{31}(a) - 8970411302925107200\, \cos^{29}(a) + 3172462533961318400\, \cos^{27}(a) - 927945291183685632\, \cos^{25}(a) + 223063771919155200\, \cos^{23}(a) - 43680444501196800\, \cos^{21}(a) + 6886556565504000\, \cos^{19}(a) - 860819570688000\, \cos^{17}(a) + 83622472581120\, \cos^{15}(a) - 6148711219200\, \cos^{13}(a) + 330302668800\, \cos^{11}(a) - 12341472000\, \cos^{9}(a) + 298584000\, \cos^{7}(a) - 4180176\, \cos^{5}(a) + 27720\, \cos^{3}(a) - 55\, \cos(a)$ $\cos(56a) = 36028797018963968\, \cos^{56}(a) - 504403158265495552\, \cos^{54}(a) + 3341670923508908032\, \cos^{52}(a) - 13934137247084314624\, \cos^{50}(a) + 41032014729995878400\, \cos^{48}(a) - 90753161991049707520\, \cos^{46}(a) + 156549204434560745472\, \cos^{44}(a) - 215882722150283476992\, \cos^{42}(a) + 242024770535669366784\, \cos^{40}(a) - 223143405458418565120\, \cos^{38}(a) + 170510667431813316608\, \cos^{36}(a) - 108506788365699383296\, \cos^{34}(a) + 57644231319277797376\, \cos^{32}(a) - 25573827132702851072\, \cos^{30}(a) + 9459706464902840320\, \cos^{28}(a) - 2907129303848189952\, \cos^{26}(a) + 738138299805204480\, \cos^{24}(a) - 153639646113300480\, \cos^{22}(a) + 25943536734044160\, \cos^{20}(a) - 3505883342438400\, \cos^{18}(a) + 372500105134080\, \cos^{16}(a) - 30408171847680\, \cos^{14}(a) + 1849694945280\, \cos^{12}(a) - 80421519360\, \cos^{10}(a) + 2356099200\, \cos^{8}(a) - 42561792\, \cos^{6}(a) + 409248\, \cos^{4}(a) - 1568\, \cos^{2}(a) + 1$ $\cos(57a) = 72057594037927936\, \cos^{57}(a) - 1026820715040473088\, \cos^{55}(a) + 6931039826523193344\, \cos^{53}(a) - 29478311360953581568\, \cos^{51}(a) + 88643507040603340800\, \cos^{49}(a) - 200470700537979863040\, \cos^{47}(a) + 354099390982935019520\, \cos^{45}(a) - 500797710104436670464\, \cos^{43}(a) + 576811648245288665088\, \cos^{41}(a) - 547436980973537853440\, \cos^{39}(a) + 431543407341905903616\, \cos^{37}(a) - 283999910365325426688\, \cos^{35}(a) + 156462913580896878592\, \cos^{33}(a) - 72213652421952405504\, \cos^{31}(a) + 27889824232730787840\, \cos^{29}(a) - 8986721141657698304\, \cos^{27}(a) + 2404221890794094592\, \cos^{25}(a) - 530343064145756160\, \cos^{23}(a) + 95567517969285120\, \cos^{21}(a) - 13898323250380800\, \cos^{19}(a) + 1605819780956160\, \cos^{17}(a) - 144438816276480\, \cos^{15}(a) + 9848101109760\, \cos^{13}(a) - 491145707520\, \cos^{11}(a) + 17053670400\, \cos^{9}(a) - 383707584\, \cos^{7}(a) + 4998672\, \cos^{5}(a) - 30856\, \cos^{3}(a) + 57\, \cos(a)$ $\cos(58a) = 144115188075855872\, \cos^{58}(a) - 2089670227099910144\, \cos^{56}(a) + 14366482811311882240\, \cos^{54}(a) - 62298293645416071168\, \cos^{52}(a) + 191221151328290996224\, \cos^{50}(a) - 441973415805955604480\, \cos^{48}(a) + 798951943956919746560\, \cos^{46}(a) - 1158144624643434086400\, \cos^{44}(a) + 1369506018640860807168\, \cos^{42}(a) - 1336898732482745073664\, \cos^{40}(a) + 1086230220142230372352\, \cos^{38}(a) - 738510488162464169984\, \cos^{36}(a) + 421432615527493140480\, \cos^{34}(a) - 202071536163182608384\, \cos^{32}(a) + 81353475598164426752\, \cos^{30}(a) - 27433148748218236928\, \cos^{28}(a) + 7715573085436379136\, \cos^{26}(a) - 1798824428096716800\, \cos^{24}(a) + 344774682051870720\, \cos^{22}(a) - 53740183234805760\, \cos^{20}(a) + 6717522904350720\, \cos^{18}(a) - 661377737687040\, \cos^{16}(a) + 50104374067200\, \cos^{14}(a) - 2831986360320\, \cos^{12}(a) + 114528860160\, \cos^{10}(a) - 3123514368\, \cos^{8}(a) + 52559136\, \cos^{6}(a) - 470960\, \cos^{4}(a) + 1682\, \cos^{2}(a) - 1$ $\cos(59a) = 288230376151711744\, \cos^{59}(a) - 4251398048237748224\, \cos^{57}(a) + 29759786337664237568\, \cos^{55}(a) - 131527627117355335680\, \cos^{53}(a) + 411920614017535574016\, \cos^{51}(a) - 972590338652514549760\, \cos^{49}(a) + 1798374588451819356160\, \cos^{47}(a) - 2670388640269803192320\, \cos^{45}(a) + 3239809747386158284800\, \cos^{43}(a) - 3250609113210778812416\, \cos^{41}(a) + 2719897421257998598144\, \cos^{39}(a) - 1908564383666834243584\, \cos^{37}(a) + 1126865141420311707648\, \cos^{35}(a) - 560605985907262095360\, \cos^{33}(a) + 234920603618281259008\, \cos^{31}(a) - 82756121729167261696\, \cos^{29}(a) + 24417867312530456576\, \cos^{27}(a) - 6001870746987528192\, \cos^{25}(a) + 1219892428249497600\, \cos^{23}(a) - 203047884438896640\, \cos^{21}(a) + 27333369059082240\, \cos^{19}(a) - 2928575256330240\, \cos^{17}(a) + 244647564410880\, \cos^{15}(a) - 15512073830400\, \cos^{13}(a) + 720203427840\, \cos^{11}(a) - 23300699136\, \cos^{9}(a) + 488825856\, \cos^{7}(a) - 5940592\, \cos^{5}(a) + 34220\, \cos^{3}(a) - 59\, \cos(a)$ $\cos(60a) = 576460752303423488\, \cos^{60}(a) - 8646911284551352320\, \cos^{58}(a) + 61609242902428385280\, \cos^{56}(a) - 277421737046022553600\, \cos^{54}(a) + 886139521680487219200\, \cos^{52}(a) - 2136401828633320095744\, \cos^{50}(a) + 4038722592709594316800\, \cos^{48}(a) - 6139729224496526131200\, \cos^{46}(a) + 7637764119415750656000\, \cos^{44}(a) - 7870724245062418432000\, \cos^{42}(a) + 6776693574998742269952\, \cos^{40}(a) - 4903358987475898859520\, \cos^{38}(a) + 2992240771003087585280\, \cos^{36}(a) - 1542644587342017331200\, \cos^{34}(a) + 671912743399745126400\, \cos^{32}(a) - 246865719056498950144\, \cos^{30}(a) + 76268883373279150080\, \cos^{28}(a) - 19719314579411435520\, \cos^{26}(a) + 4238609284595712000\, \cos^{24}(a) - 750870450929664000\, \cos^{22}(a) + 108406921352970240\, \cos^{20}(a) - 12574673417011200\, \cos^{18}(a) + 1150672866508800\, \cos^{16}(a) - 81128521728000\, \cos^{14}(a) + 4272393216000\, \cos^{12}(a) - 161130258432\, \cos^{10}(a) + 4101166080\, \cos^{8}(a) - 64440320\, \cos^{6}(a) + 539400\, \cos^{4}(a) - 1800\, \cos^{2}(a) + 1$ $\cos(61a) = 1152921504606846976\, \cos^{61}(a) - 17582052945254416384\, \cos^{59}(a) + 127469883853094518784\, \cos^{57}(a) - 584603260429709344768\, \cos^{55}(a) + 1903806670478329774080\, \cos^{53}(a) - 4684724271284175765504\, \cos^{51}(a) + 9050035524071703183360\, \cos^{49}(a) - 14077833037444871618560\, \cos^{47}(a) + 17945916879101304504320\, \cos^{45}(a) - 18981258237510995148800\, \cos^{43}(a) + 16803996263208263352320\, \cos^{41}(a) - 12526615396209796317184\, \cos^{39}(a) + 7893045925673009414144\, \cos^{37}(a) - 4212154316104346370048\, \cos^{35}(a) + 1904431472706752348160\, \cos^{33}(a) - 728652041731279159296\, \cos^{31}(a) + 235293888475725561856\, \cos^{29}(a) - 63856496471353327616\, \cos^{27}(a) + 14479089316178952192\, \cos^{25}(a) - 2721633330108825600\, \cos^{23}(a) + 419861727144837120\, \cos^{21}(a) - 52482715893104640\, \cos^{19}(a) + 5229920989347840\, \cos^{17}(a) - 406904607866880\, \cos^{15}(a) + 24056860262400\, \cos^{13}(a) - 1042463944704\, \cos^{11}(a) + 31503031296\, \cos^{9}(a) - 617706496\, \cos^{7}(a) + 7019392\, \cos^{5}(a) - 37820\, \cos^{3}(a) + 61\, \cos(a)$ $\cos(62a) = 2305843009213693952\, \cos^{62}(a) - 35740566642812256256\, \cos^{60}(a) + 263586678990740389888\, \cos^{58}(a) - 1230815763761847074816\, \cos^{56}(a) + 4085035078002682101760\, \cos^{54}(a) - 10255588064248838750208\, \cos^{52}(a) + 20236472876776726462464\, \cos^{50}(a) - 32194388667599337553920\, \cos^{48}(a) + 42031562982699135139840\, \cos^{46}(a) - 45600280594437740953600\, \cos^{44}(a) + 41478716771478945136640\, \cos^{42}(a) - 31829924367418334904320\, \cos^{40}(a) + 20689450838821917687808\, \cos^{38}(a) - 11416549403211780325376\, \cos^{36}(a) + 5351507532755522027520\, \cos^{34}(a) - 2129216826862303444992\, \cos^{32}(a) + 717453496007950073856\, \cos^{30}(a) - 203981876315985805312\, \cos^{28}(a) + 48677493211769339904\, \cos^{26}(a) - 9681875944813363200\, \cos^{24}(a) + 1590593905219338240\, \cos^{22}(a) - 213372353139179520\, \cos^{20}(a) + 23034515395706880\, \cos^{18}(a) - 1964482082242560\, \cos^{16}(a) + 129242242252800\, \cos^{14}(a) - 6357321105408\, \cos^{12}(a) + 224136321024\, \cos^{10}(a) - 5336579072\, \cos^{8}(a) + 78479104\, \cos^{6}(a) - 615040\, \cos^{4}(a) + 1922\, \cos^{2}(a) - 1$ $\cos(63a) = 4611686018427387904\, \cos^{63}(a) - 72634054790231359488\, \cos^{61}(a) + 544755410926735196160\, \cos^{59}(a) - 2589101411376788668416\, \cos^{57}(a) + 8754673416435073548288\, \cos^{55}(a) - 22414982798976007274496\, \cos^{53}(a) + 45157670024837628690432\, \cos^{51}(a) - 73438812859270378291200\, \cos^{49}(a) + 98140959002843141898240\, \cos^{47}(a) - 109146478067976786411520\, \cos^{45}(a) + 101938691780468885422080\, \cos^{43}(a) - 80463844998044933160960\, \cos^{41}(a) + 53905517073853631692800\, \cos^{39}(a) - 30726144732096570064896\, \cos^{37}(a) + 14915169381615390425088\, \cos^{35}(a) - 6162865126431359238144\, \cos^{33}(a) + 2163559033747179307008\, \cos^{31}(a) - 643257641107697172480\, \cos^{29}(a) + 161211482894892007424\, \cos^{27}(a) - 33842841205805678592\, \cos^{25}(a) + 5902821140547502080\, \cos^{23}(a) - 846606433423196160\, \cos^{21}(a) + 98551746684518400\, \cos^{19}(a) - 9158885153832960\, \cos^{17}(a) + 665389092372480\, \cos^{15}(a) - 36771502473216\, \cos^{13}(a) + 1490736586752\, \cos^{11}(a) - 42176189440\, \cos^{9}(a) + 774664704\, \cos^{7}(a) - 8249472\, \cos^{5}(a) + 41664\, \cos^{3}(a) - 63\, \cos(a)$ $\cos(64a) = 9223372036854775808\, \cos^{64}(a) - 147573952589676412928\, \cos^{62}(a) + 1125251388496282648576\, \cos^{60}(a) - 5441789501744317726720\, \cos^{58}(a) + 18740162596631994171392\, \cos^{56}(a) - 48915000675954696650752\, \cos^{54}(a) + 100570928113924096131072\, \cos^{52}(a) - 167114098595317483044864\, \cos^{50}(a) + 228476306673285621350400\, \cos^{48}(a) - 260324519118652707962880\, \cos^{46}(a) + 249477664155375511797760\, \cos^{44}(a) - 202406406767568811458560\, \cos^{42}(a) + 139640958515125598289920\, \cos^{40}(a) - 82141740303015057817600\, \cos^{38}(a) + 41246888166442561175552\, \cos^{36}(a) - 17677237785618240503808\, \cos^{34}(a) + 6456334894356662059008\, \cos^{32}(a) - 2003968778223344418816\, \cos^{30}(a) + 526404842105769820160\, \cos^{28}(a) - 116363175623380697088\, \cos^{26}(a) + 21487518225908367360\, \cos^{24}(a) - 3283806772065730560\, \cos^{22}(a) + 410475846508216320\, \cos^{20}(a) - 41352285703372800\, \cos^{18}(a) + 3295260266987520\, \cos^{16}(a) - 202785247199232\, \cos^{14}(a) + 9338794278912\, \cos^{12}(a) - 308488699904\, \cos^{10}(a) + 6885908480\, \cos^{8}(a) - 94978048\, \cos^{6}(a) + 698368\, \cos^{4}(a) - 2048\, \cos^{2}(a) + 1$ $\cos(65a) = 18446744073709551616\, \cos^{65}(a) - 299759591197780213760\, \cos^{63}(a) + 2323136831782796656640\, \cos^{61}(a) - 11428334414415370649600\, \cos^{59}(a) + 40069426604640777011200\, \cos^{57}(a) - 106584674768344466849792\, \cos^{55}(a) + 223556839026824199536640\, \cos^{53}(a) - 379385867215472594780160\, \cos^{51}(a) + 530391426205841620992000\, \cos^{49}(a) - 618789997240148557824000\, \cos^{47}(a) + 608101806378727810007040\, \cos^{45}(a) - 506751505315606508339200\, \cos^{43}(a) + 359745762028296129740800\, \cos^{41}(a) - 218188997679883747328000\, \cos^{39}(a) + 113219921064981692416000\, \cos^{37}(a) - 50269644952851871432704\, \cos^{35}(a) + 19075534915144683356160\, \cos^{33}(a) - 6171496590193868144640\, \cos^{31}(a) + 1696067325319236812800\, \cos^{29}(a) - 393937834141653401600\, \cos^{27}(a) + 76817877657622413312\, \cos^{25}(a) - 12470434684678963200\, \cos^{23}(a) + 1667558126439628800\, \cos^{21}(a) - 181256318091264000\, \cos^{19}(a) + 15749405687808000\, \cos^{17}(a) - 1070959586770944\, \cos^{15}(a) + 55449091031040\, \cos^{13}(a) - 2107713986560\, \cos^{11}(a) + 55948006400\, \cos^{9}(a) - 964620800\, \cos^{7}(a) + 9646208\, \cos^{5}(a) - 45760\, \cos^{3}(a) + 65\, \cos(a)$ $\cos(66a) = 36893488147419103232\, \cos^{66}(a) - 608742554432415203328\, \cos^{64}(a) + 4793847616155269726208\, \cos^{62}(a) - 23981920217327023947776\, \cos^{60}(a) + 85580642711025871749120\, \cos^{58}(a) - 231909512133320927870976\, \cos^{56}(a) + 496028678729603095724032\, \cos^{54}(a) - 859342662544869285691392\, \cos^{52}(a) + 1227896951007000725028864\, \cos^{50}(a) - 1466056301153582736998400\, \cos^{48}(a) + 1476528131876108327976960\, \cos^{46}(a) - 1262980674786588528476160\, \cos^{44}(a) + 921897930824161070940160\, \cos^{42}(a) - 576018953874893092945920\, \cos^{40}(a) + 308581582432978442649600\, \cos^{38}(a) - 141786178072146304040960\, \cos^{36}(a) + 55828307615907607216128\, \cos^{34}(a) - 18799328074744398348288\, \cos^{32}(a) + 5396103428861818044416\, \cos^{30}(a) - 1314280510389076623360\, \cos^{28}(a) + 269998930938625523712\, \cos^{26}(a) - 46428387595266293760\, \cos^{24}(a) + 6618923024944988160\, \cos^{22}(a) - 772988482690744320\, \cos^{20}(a) + 72851097078988800\, \cos^{18}(a) - 5437179440529408\, \cos^{16}(a) + 313683429261312\, \cos^{14}(a) - 13554222252032\, \cos^{12}(a) + 420384712704\, \cos^{10}(a) - 8815150080\, \cos^{8}(a) + 114270464\, \cos^{6}(a) - 789888\, \cos^{4}(a) + 2178\, \cos^{2}(a) - 1$ $\cos(67a) = 73786976294838206464\, \cos^{67}(a) - 1235931852938539958272\, \cos^{65}(a) + 9887454823508319666176\, \cos^{63}(a) - 50286977266436844552192\, \cos^{61}(a) + 182589619836467114147840\, \cos^{59}(a) - 503888450871282632753152\, \cos^{57}(a) + 1098642032227550658297856\, \cos^{55}(a) - 1942242164116562770919424\, \cos^{53}(a) + 2835179769229474044837888\, \cos^{51}(a) - 3462504028513007094988800\, \cos^{49}(a) + 3571846260992365213777920\, \cos^{47}(a) - 3134063155951904866959360\, \cos^{45}(a) + 2350547366963928650219520\, \cos^{43}(a) - 1511783669778082315632640\, \cos^{41}(a) + 835352162545840632627200\, \cos^{39}(a) - 396792277209274300497920\, \cos^{37}(a) + 161926260184667085864960\, \cos^{35}(a) - 56674191064633480052736\, \cos^{33}(a) + 16963703447917504233472\, \cos^{31}(a) - 4324628346097390059520\, \cos^{29}(a) + 933935696018904449024\, \cos^{27}(a) - 169674652848155000832\, \cos^{25}(a) + 25708280734568939520\, \cos^{23}(a) - 3213535091821117440\, \cos^{21}(a) + 326958512249241600\, \cos^{19}(a) - 26623764568866816\, \cos^{17}(a) + 1698326445293568\, \cos^{15}(a) - 82557535535104\, \cos^{13}(a) + 2948483411968\, \cos^{11}(a) - 73578306560\, \cos^{9}(a) + 1193161728\, \cos^{7}(a) - 11225984\, \cos^{5}(a) + 50116\, \cos^{3}(a) - 67\, \cos(a)$ $\cos(68a) = 147573952589676412928\, \cos^{68}(a) - 2508757194024499019776\, \cos^{66}(a) + 20383652201449054535680\, \cos^{64}(a) - 105367802149028958830592\, \cos^{62}(a) + 389161159890261252243456\, \cos^{60}(a) - 1093357544453591137255424\, \cos^{58}(a) + 2429193576588422244466688\, \cos^{56}(a) - 4380513006962728637562880\, \cos^{54}(a) + 6529702201003817375367168\, \cos^{52}(a) - 8152905008033014915006464\, \cos^{50}(a) + 8609748823138313164554240\, \cos^{48}(a) - 7744654443779918061895680\, \cos^{46}(a) + 5964075408714445828915200\, \cos^{44}(a) - 3945465270380325702205440\, \cos^{42}(a) + 2246723278966574358200320\, \cos^{40}(a) - 1102166136851527043645440\, \cos^{38}(a) + 465638698441480475770880\, \cos^{36}(a) - 169176689745174567321600\, \cos^{34}(a) + 52726734970579406815232\, \cos^{32}(a) - 14045360121056598163456\, \cos^{30}(a) + 3182151902426885521408\, \cos^{28}(a) - 609348236634935525376\, \cos^{26}(a) + 97844949064404172800\, \cos^{24}(a) - 13045993208587223040\, \cos^{22}(a) + 1426905507189227520\, \cos^{20}(a) - 126098626216722432\, \cos^{18}(a) + 8833832331116544\, \cos^{16}(a) - 478798500331520\, \cos^{14}(a) + 19451189075968\, \cos^{12}(a) - 567541325824\, \cos^{10}(a) + 11201473536\, \cos^{8}(a) - 136722432\, \cos^{6}(a) + 890120\, \cos^{4}(a) - 2312\, \cos^{2}(a) + 1$ $\cos(69a) = 295147905179352825856\, \cos^{69}(a) - 5091301364343836246016\, \cos^{67}(a) + 42003236255836649029632\, \cos^{65}(a) - 220623059121566237327360\, \cos^{63}(a) + 828609297046959349039104\, \cos^{61}(a) - 2369304708743649388658688\, \cos^{59}(a) + 5362275604048127121686528\, \cos^{57}(a) - 9859668046153007933423616\, \cos^{55}(a) + 15001646566124197521653760\, \cos^{53}(a) - 19140989785295503874850816\, \cos^{51}(a) + 20682001674789633424097280\, \cos^{49}(a) - 19061155148552201337569280\, \cos^{47}(a) + 15062213973380796524789760\, \cos^{45}(a) - 10241477907724580054630400\, \cos^{43}(a) + 6005230227711231032033280\, \cos^{41}(a) - 3039684436248894719918080\, \cos^{39}(a) + 1328069674092235252039680\, \cos^{37}(a) - 500279639675016220508160\, \cos^{35}(a) + 162127661005792293683200\, \cos^{33}(a) - 45054423690030700560384\, \cos^{31}(a) + 10688932150951161102336\, \cos^{29}(a) - 2152632169288775499776\, \cos^{27}(a) + 365364550976963346432\, \cos^{25}(a) - 51800267151743385600\, \cos^{23}(a) + 6067346106199572480\, \cos^{21}(a) - 579155764682686464\, \cos^{19}(a) + 44291429231099904\, \cos^{17}(a) - 2655923445956608\, \cos^{15}(a) + 121459913687040\, \cos^{13}(a) - 4083566063616\, \cos^{11}(a) + 95981253632\, \cos^{9}(a) - 1466606592\, \cos^{7}(a) + 13006224\, \cos^{5}(a) - 54740\, \cos^{3}(a) + 69\, \cos(a)$ $\cos(70a) = 590295810358705651712\, \cos^{70}(a) - 10330176681277348904960\, \cos^{68}(a) + 86515229705697797079040\, \cos^{66}(a) - 461629770444581529190400\, \cos^{64}(a) + 1762586396242947656908800\, \cos^{62}(a) - 5127770577377560029560832\, \cos^{60}(a) + 11817908752549845380628480\, \cos^{58}(a) - 22148529668894438111313920\, \cos^{56}(a) + 34383806139211123680870400\, \cos^{54}(a) - 44811681771594825125068800\, \cos^{52}(a) + 49516908357612281763201024\, \cos^{50}(a) - 46732059120242715839692800\, \cos^{48}(a) + 37869082390541511111475200\, \cos^{46}(a) - 26447031224163605938176000\, \cos^{44}(a) + 15955925725802787766272000\, \cos^{42}(a) - 8326092151464363798036480\, \cos^{40}(a) + 3758305485035997547724800\, \cos^{38}(a) - 1466197977791512916787200\, \cos^{36}(a) + 493432011756759154688000\, \cos^{34}(a) - 142835582350640807936000\, \cos^{32}(a) + 35423224422958920368128\, \cos^{30}(a) - 7487416241004436520960\, \cos^{28}(a) + 1340077338588862218240\, \cos^{26}(a) - 201445483367890944000\, \cos^{24}(a) + 25180685420986368000\, \cos^{22}(a) - 2585217036554600448\, \cos^{20}(a) + 214681484678922240\, \cos^{18}(a) - 14145679223029760\, \cos^{16}(a) + 721718327705600\, \cos^{14}(a) - 27618321203200\, \cos^{12}(a) + 759503833088\, \cos^{10}(a) - 14134686720\, \cos^{8}(a) + 162734880\, \cos^{6}(a) - 999600\, \cos^{4}(a) + 2450\, \cos^{2}(a) - 1$ $\cos(71a) = 1180591620717411303424\, \cos^{71}(a) - 20955501267734050635776\, \cos^{69}(a) + 178121760775739430404096\, \cos^{67}(a) - 965262777144999707410432\, \cos^{65}(a) + 3745795851607461551144960\, \cos^{63}(a) - 11084150451802079408160768\, \cos^{61}(a) + 26005122213843340149915648\, \cos^{59}(a) - 49659334941837003344314368\, \cos^{57}(a) + 78627280324575255295164416\, \cos^{55}(a) - 104625010109313847771791360\, \cos^{53}(a) + 118174806500520067401252864\, \cos^{51}(a) - 114146119915275065103482880\, \cos^{49}(a) + 94799319929635223560519680\, \cos^{47}(a) - 67956276421708008401141760\, \cos^{45}(a) + 42153329359330155587174400\, \cos^{43}(a) - 22657414530639958628106240\, \cos^{41}(a) + 10556295406320889815367680\, \cos^{39}(a) - 4260465629675261085614080\, \cos^{37}(a) + 1487143663188534529884160\, \cos^{35}(a) - 447798825707073909555200\, \cos^{33}(a) + 115900872535948541296640\, \cos^{31}(a) - 25663764632960034144256\, \cos^{29}(a) + 4832786846466499936256\, \cos^{27}(a) - 768255517712745234432\, \cos^{25}(a) + 102161637993716121600\, \cos^{23}(a) - 11237780179308773376\, \cos^{21}(a) + 1008518734040530944\, \cos^{19}(a) - 72582787677159424\, \cos^{17}(a) + 4099360101367808\, \cos^{15}(a) - 176696556093440\, \cos^{13}(a) + 5602573729792\, \cos^{11}(a) - 124250627072\, \cos^{9}(a) + 1792076352\, \cos^{7}(a) - 15005424\, \cos^{5}(a) + 59640\, \cos^{3}(a) - 71\, \cos(a)$ $\cos(72a) = 2361183241434822606848\, \cos^{72}(a) - 42501298345826806923264\, \cos^{70}(a) + 366573698232756209713152\, \cos^{68}(a) - 2017040783995697211899904\, \cos^{66}(a) + 7953221473659504631480320\, \cos^{64}(a) - 23930887299847106473230336\, \cos^{62}(a) + 57138015005064240329392128\, \cos^{60}(a) - 111136578636223852069257216\, \cos^{58}(a) + 179403090318044948701642752\, \cos^{56}(a) - 243633826357838819224453120\, \cos^{54}(a) + 281161294772634959927574528\, \cos^{52}(a) - 277809148188162411970166784\, \cos^{50}(a) + 236330698979513162960732160\, \cos^{48}(a) - 173781635233957527913758720\, \cos^{46}(a) + 110753689942823917112524800\, \cos^{44}(a) - 61270754787082705022484480\, \cos^{42}(a) + 29438682964106143428771840\, \cos^{40}(a) - 12279236744386519718952960\, \cos^{38}(a) + 4440485304168581976555520\, \cos^{36}(a) - 1389029663170906973798400\, \cos^{34}(a) + 374637327422537890529280\, \cos^{32}(a) - 86750753688878988656640\, \cos^{30}(a) + 17152989933937436393472\, \cos^{28}(a) - 2876588374014352687104\, \cos^{26}(a) + 405768759355323187200\, \cos^{24}(a) - 47656245779603914752\, \cos^{22}(a) + 4602254504635662336\, \cos^{20}(a) - 359847060033241088\, \cos^{18}(a) + 22344399425765376\, \cos^{16}(a) - 1075111439892480\, \cos^{14}(a) + 38823468662784\, \cos^{12}(a) - 1008005087232\, \cos^{10}(a) + 17718839424\, \cos^{8}(a) - 192745728\, \cos^{6}(a) + 1118880\, \cos^{4}(a) - 2592\, \cos^{2}(a) + 1$ $\cos(73a) = 4722366482869645213696\, \cos^{73}(a) - 86183188312371025149952\, \cos^{71}(a) + 754102897733246470062080\, \cos^{69}(a) - 4212203328767133854203904\, \cos^{67}(a) + 16871705724464008970371072\, \cos^{65}(a) - 51607570451301674497605632\, \cos^{63}(a) + 125360180461930560066945024\, \cos^{61}(a) - 248278279486291044288430080\, \cos^{59}(a) + 408465515577926900747599872\, \cos^{57}(a) - 565894933040252893744070656\, \cos^{55}(a) + 666947599654583767626940416\, \cos^{53}(a) - 673793102876844891341586432\, \cos^{51}(a) + 586807517874301391024947200\, \cos^{49}(a) - 442362590397550279388037120\, \cos^{47}(a) + 289463656307355842626191360\, \cos^{45}(a) - 164694838933495565632143360\, \cos^{43}(a) + 81534780458852245485649920\, \cos^{41}(a) - 35114768895093929253273600\, \cos^{39}(a) + 13141436238012425038725120\, \cos^{37}(a) - 4265202989530348477480960\, \cos^{35}(a) + 1197073480552149690613760\, \cos^{33}(a) - 289402379913706518609920\, \cos^{31}(a) + 59969744500834906931200\, \cos^{29}(a) - 10585963594495205310464\, \cos^{27}(a) + 1579793036423391608832\, \cos^{25}(a) - 197474129552923951104\, \cos^{23}(a) + 20442289188580098048\, \cos^{21}(a) - 1728212854107013120\, \cos^{19}(a) + 117271586528690176\, \cos^{17}(a) - 6249582981152768\, \cos^{15}(a) + 254343493419008\, \cos^{13}(a) - 7618583904256\, \cos^{11}(a) + 159688305920\, \cos^{9}(a) - 2177567808\, \cos^{7}(a) + 17243184\, \cos^{5}(a) - 64824\, \cos^{3}(a) + 73\, \cos(a)$ $\cos(74a) = 9444732965739290427392\, \cos^{74}(a) - 174727559866176872906752\, \cos^{72}(a) + 1550707093812319747047424\, \cos^{70}(a) - 8790980355767023918120960\, \cos^{68}(a) + 35760452232923715152642048\, \cos^{66}(a) - 111168362376262853626691584\, \cos^{64}(a) + 274651248223708226607120384\, \cos^{62}(a) - 553694573977646328906252288\, \cos^{60}(a) + 928067609792077653564456960\, \cos^{58}(a) - 1311192956398550736189784064\, \cos^{56}(a) + 1577529025667006354478333952\, \cos^{54}(a) - 1628747500526324742610747392\, \cos^{52}(a) + 1451424183936765194020061184\, \cos^{50}(a) - 1121055879774613721736806400\, \cos^{48}(a) + 752708947848669213166141440\, \cos^{46}(a) - 440143367809815048376811520\, \cos^{44}(a) + 224340315704787195993784320\, \cos^{42}(a) - 99668220754294001935319040\, \cos^{40}(a) + 38562109220411369796403200\, \cos^{38}(a) - 12970891283229278931517440\, \cos^{36}(a) + 3783176624275206355025920\, \cos^{34}(a) - 953442087249950927749120\, \cos^{32}(a) + 206690242690548802519040\, \cos^{30}(a) - 38324917122927847014400\, \cos^{28}(a) + 6036174446861135904768\, \cos^{26}(a) - 800717018461171089408\, \cos^{24}(a) + 88540824156764110848\, \cos^{22}(a) - 8058680212849688576\, \cos^{20}(a) + 594390233090621440\, \cos^{18}(a) - 34843565388070912\, \cos^{16}(a) + 1583798426730496\, \cos^{14}(a) - 54060636471296\, \cos^{12}(a) + 1327381699072\, \cos^{10}(a) - 22073975040\, \cos^{8}(a) + 227232096\, \cos^{6}(a) - 1248528\, \cos^{4}(a) + 2738\, \cos^{2}(a) - 1$ $\cos(75a) = 18889465931478580854784\, \cos^{75}(a) - 354177486215223391027200\, \cos^{73}(a) + 3187597375937010519244800\, \cos^{71}(a) - 18336063609267294306304000\, \cos^{69}(a) + 75733107794614564159488000\, \cos^{67}(a) - 239208430476989716223754240\, \cos^{65}(a) + 600910066898718127711846400\, \cos^{63}(a) - 1232749328417223217879449600\, \cos^{61}(a) + 2104413499070446351417344000\, \cos^{59}(a) - 3030851428375028373127168000\, \cos^{57}(a) + 3720952984374265602700738560\, \cos^{55}(a) - 3924442600707233252848435200\, \cos^{53}(a) + 3576641470750375279381708800\, \cos^{51}(a) - 2828919277423528834498560000\, \cos^{49}(a) + 1947780486094888705720320000\, \cos^{47}(a) - 1169750391926985939379814400\, \cos^{45}(a) + 613375470343069957619712000\, \cos^{43}(a) - 280871221967440249356288000\, \cos^{41}(a) + 112238987335916668846080000\, \cos^{39}(a) - 39083218804470982901760000\, \cos^{37}(a) + 11831556238080761187532800\, \cos^{35}(a) - 3103957655052051546112000\, \cos^{33}(a) + 702782865294804123648000\, \cos^{31}(a) - 136619578746690600960000\, \cos^{29}(a) + 22658312488217477120000\, \cos^{27}(a) - 3181227073345733787648\, \cos^{25}(a) + 374555777866452172800\, \cos^{23}(a) - 36559649614279475200\, \cos^{21}(a) + 2916993320288256000\, \cos^{19}(a) - 186958717304832000\, \cos^{17}(a) + 9417179834613760\, \cos^{15}(a) - 362464766361600\, \cos^{13}(a) + 10273347302400\, \cos^{11}(a) - 203836256000\, \cos^{9}(a) + 2632032000\, \cos^{7}(a) - 19740240\, \cos^{5}(a) + 70300\, \cos^{3}(a) - 75\, \cos(a)$ $\cos(76a) = 37778931862957161709568\, \cos^{76}(a) - 717799705396186072481792\, \cos^{74}(a) + 6549922311740197911396352\, \cos^{72}(a) - 38222834312346908359655424\, \cos^{70}(a) + 160257195944996152237096960\, \cos^{68}(a) - 514177313186903147600150528\, \cos^{66}(a) + 1312988496173699109050384384\, \cos^{64}(a) - 2740149905058154662366019584\, \cos^{62}(a) + 4762521572118539031740940288\, \cos^{60}(a) - 6989770466542134399818792960\, \cos^{58}(a) + 8753098925147081941591261184\, \cos^{56}(a) - 9426414227081472860175204352\, \cos^{54}(a) + 8782030442027075301374164992\, \cos^{52}(a) - 7109262738783822863017181184\, \cos^{50}(a) + 5016616851964391133177446400\, \cos^{48}(a) - 3092209731702641091925770240\, \cos^{46}(a) + 1666894308495954963616235520\, \cos^{44}(a) - 786082759639667694706360320\, \cos^{42}(a) + 324146195426127339627479040\, \cos^{40}(a) - 116728546829353335599923200\, \cos^{38}(a) + 36634003759390801306583040\, \cos^{36}(a) - 9991091934379309447249920\, \cos^{34}(a) + 2359007817839559175045120\, \cos^{32}(a) - 479929400183930004439040\, \cos^{30}(a) + 83641542099362801254400\, \cos^{28}(a) - 12398628593552603480064\, \cos^{26}(a) + 1549828574194075435008\, \cos^{24}(a) - 161660123385323061248\, \cos^{22}(a) + 13892666853426200576\, \cos^{20}(a) - 968307667700285440\, \cos^{18}(a) + 53677925057298432\, \cos^{16}(a) - 2308727959453696\, \cos^{14}(a) + 74607331076096\, \cos^{12}(a) - 1735054211072\, \cos^{10}(a) + 27338039040\, \cos^{8}(a) - 266712576\, \cos^{6}(a) + 1389128\, \cos^{4}(a) - 2888\, \cos^{2}(a) + 1$ $\cos(77a) = 75557863725914323419136\, \cos^{77}(a) - 1454488876723850725818368\, \cos^{75}(a) + 13454022109695619213819904\, \cos^{73}(a) - 79633266000630827238555648\, \cos^{71}(a) + 338850455499259598780497920\, \cos^{69}(a) - 1104087734168420859359789056\, \cos^{67}(a) + 2865185422824387934324523008\, \cos^{65}(a) - 6081209877015027452443885568\, \cos^{63}(a) + 10757792472654301281361330176\, \cos^{61}(a) - 16083954432154715151054929920\, \cos^{59}(a) + 20537049278669192256309690368\, \cos^{57}(a) - 22573781438537211323051147264\, \cos^{55}(a) + 21488503484761383855596765184\, \cos^{53}(a) - 17795166948318021005416071168\, \cos^{51}(a) + 12862152981352311100853452800\, \cos^{49}(a) - 8132199949500170889571860480\, \cos^{47}(a) + 4503539008918895866612285440\, \cos^{45}(a) - 2185540989622405347032432640\, \cos^{43}(a) + 929163612819694928611246080\, \cos^{41}(a) - 345696080994623340045926400\, \cos^{39}(a) + 112351226323252585514926080\, \cos^{37}(a) - 31813740106839380082032640\, \cos^{35}(a) + 7821973290731169896202240\, \cos^{33}(a) - 1662641665662664132526080\, \cos^{31}(a) + 303902662945416203468800\, \cos^{29}(a) - 47455569675322684080128\, \cos^{27}(a) + 6280884221733884657664\, \cos^{25}(a) - 697876024637098295296\, \cos^{23}(a) + 64344983321131876352\, \cos^{21}(a) - 4853608655688826880\, \cos^{19}(a) + 294314567419428864\, \cos^{17}(a) - 14034635753521152\, \cos^{15}(a) + 511679428513792\, \cos^{13}(a) - 13743455724544\, \cos^{11}(a) + 258512334080\, \cos^{9}(a) - 3165457152\, \cos^{7}(a) + 22518496\, \cos^{5}(a) - 76076\, \cos^{3}(a) + 77\, \cos(a)$ $\cos(78a) = 151115727451828646838272\, \cos^{78}(a) - 2946756685310658613346304\, \cos^{76}(a) + 27625843924787424500121600\, \cos^{74}(a) - 165816454313001852388507648\, \cos^{72}(a) + 715923745310866105920651264\, \cos^{70}(a) - 2368432664281837870956675072\, \cos^{68}(a) + 6244548158835679016249196544\, \cos^{66}(a) - 13475408250203754013938155520\, \cos^{64}(a) + 24255734850366757225088679936\, \cos^{62}(a) - 36930430436427969333850800128\, \cos^{60}(a) + 48063869023880518912438173696\, \cos^{58}(a) - 53900661802221504587693555712\, \cos^{56}(a) + 52403421196604240571368734720\, \cos^{54}(a) - 44372364338663117312206307328\, \cos^{52}(a) + 32833568701488445064724086784\, \cos^{50}(a) - 21281016750964732912321167360\, \cos^{48}(a) + 12099287749540432825150341120\, \cos^{46}(a) - 6037976287740765657681100800\, \cos^{44}(a) + 2644409985279057551928852480\, \cos^{42}(a) - 1015538357415374019719331840\, \cos^{40}(a) + 341430999475858506629775360\, \cos^{38}(a) - 100261483973069561470648320\, \cos^{36}(a) + 25635038515841649239654400\, \cos^{34}(a) - 5684291149164887440097280\, \cos^{32}(a) + 1087734726074762411376640\, \cos^{30}(a) - 178552681450008169414656\, \cos^{28}(a) + 24960397037020372795392\, \cos^{26}(a) - 2945580623468272025600\, \cos^{24}(a) + 290350090027586813952\, \cos^{22}(a) - 23599884164803854336\, \cos^{20}(a) + 1556936802539143168\, \cos^{18}(a) - 81747196564340736\, \cos^{16}(a) + 3332086816481280\, \cos^{14}(a) - 102094242525184\, \cos^{12}(a) + 2252078879232\, \cos^{10}(a) - 33668953344\, \cos^{8}(a) + 311749568\, \cos^{6}(a) - 1541280\, \cos^{4}(a) + 3042\, \cos^{2}(a) - 1$ $\cos(79a) = 302231454903657293676544\, \cos^{79}(a) - 5969071234347231550111744\, \cos^{77}(a) + 56706176726298699726061568\, \cos^{75}(a) - 345086930735699323990835200\, \cos^{73}(a) + 1511480756622363039079858176\, \cos^{71}(a) - 5075715784062935340693848064\, \cos^{69}(a) + 13593184051839778891858182144\, \cos^{67}(a) - 29816001923231895962200834048\, \cos^{65}(a) + 54592679577748541902621245440\, \cos^{63}(a) - 84618653345510239949062930432\, \cos^{61}(a) + 112211692479915752975931277312\, \cos^{59}(a) - 128338372883112201431696801792\, \cos^{57}(a) + 127380623831745692465788616704\, \cos^{55}(a) - 110233232162087618480009379840\, \cos^{53}(a) + 83462304351294911134864244736\, \cos^{51}(a) - 55424186483281776925495787520\, \cos^{49}(a) + 32330775448581036539872542720\, \cos^{47}(a) - 16579491584400427181974487040\, \cos^{45}(a) + 7474360960180520450890137600\, \cos^{43}(a) - 2960240327650442968049909760\, \cos^{41}(a) + 1028558079946340353305477120\, \cos^{39}(a) - 312874194269391708456222720\, \cos^{37}(a) + 83083817138522678561341440\, \cos^{35}(a) - 19190555589060944776396800\, \cos^{33}(a) + 3838111117812188955279360\, \cos^{31}(a) - 661008025845432542298112\, \cos^{29}(a) + 97376363749363429670912\, \cos^{27}(a) - 12172045468670428708864\, \cos^{25}(a) + 1278576204692271923200\, \cos^{23}(a) - 111544751650739585024\, \cos^{21}(a) + 7967482260767113216\, \cos^{19}(a) - 457808960548110336\, \cos^{17}(a) + 20698809386483712\, \cos^{15}(a) - 715867913564160\, \cos^{13}(a) + 18247613483008\, \cos^{11}(a) - 325850240768\, \cos^{9}(a) + 3788956288\, \cos^{7}(a) - 25601056\, \cos^{5}(a) + 82160\, \cos^{3}(a) - 79\, \cos(a)$ $\cos(80a) = 604462909807314587353088\, \cos^{80}(a) - 12089258196146291747061760\, \cos^{78}(a) + 116359110137908058065469440\, \cos^{76}(a) - 717799705396186072481792000\, \cos^{74}(a) + 3188777967557727930548224000\, \cos^{72}(a) - 10867355313436736787308347392\, \cos^{70}(a) + 29554800767961395654673039360\, \cos^{68}(a) - 65876552005299470940650864640\, \cos^{66}(a) + 122660767405700837819180646400\, \cos^{64}(a) - 193493041541387237123214540800\, \cos^{62}(a) + 261353815396259475285713354752\, \cos^{60}(a) - 304740614790104921775831777280\, \cos^{58}(a) + 308661909465712889519270789120\, \cos^{56}(a) - 272869885520779477531387494400\, \cos^{54}(a) + 211296973041252939581934796800\, \cos^{52}(a) - 143681941668051998915715661824\, \cos^{50}(a) + 85942567648126805992066252800\, \cos^{48}(a) - 45258270918341287189099315200\, \cos^{46}(a) + 20986698208101806559461376000\, \cos^{44}(a) - 8564890640579943488028672000\, \cos^{42}(a) + 3072654517308054726330286080\, \cos^{40}(a) - 967179388014641923542220800\, \cos^{38}(a) + 266429118250114918593331200\, \cos^{36}(a) - 64016149693963538792448000\, \cos^{34}(a) + 13360513384789265350656000\, \cos^{32}(a) - 2409750777765627495972864\, \cos^{30}(a) + 373305408948735028756480\, \cos^{28}(a) - 49304487974361230213120\, \cos^{26}(a) + 5502733032852815872000\, \cos^{24}(a) - 513439593329065984000\, \cos^{22}(a) + 39534848686338080768\, \cos^{20}(a) - 2472554723635363840\, \cos^{18}(a) + 123144815337308160\, \cos^{16}(a) - 4763822643609600\, \cos^{14}(a) + 138589469491200\, \cos^{12}(a) - 2903779360768\, \cos^{10}(a) + 41246865920\, \cos^{8}(a) - 362951680\, \cos^{6}(a) + 1705600\, \cos^{4}(a) - 3200\, \cos^{2}(a) + 1$ $\cos(81a) = 1208925819614629174706176\, \cos^{81}(a) - 24480747847196240787800064\, \cos^{79}(a) + 238687291510163347681050624\, \cos^{77}(a) - 1492305587518670844689645568\, \cos^{75}(a) + 6722642865851155185087283200\, \cos^{73}(a) - 23246191383495836613696552960\, \cos^{71}(a) + 64185317319985726650039926784\, \cos^{69}(a) - 145346288062438720773159911424\, \cos^{67}(a) + 275137536734633571600562126848\, \cos^{65}(a) - 441578762660523016149050327040\, \cos^{63}(a) + 607326284138029190520489639936\, \cos^{61}(a) - 721692922060125596527594831872\, \cos^{59}(a) + 745662191814537980470238380032\, \cos^{57}(a) - 673120394873304647528563605504\, \cos^{55}(a) + 532827178244593497643878973440\, \cos^{53}(a) - 370826187687398908966295568384\, \cos^{51}(a) + 227309321779535388909628293120\, \cos^{49}(a) - 122847317285263610918071173120\, \cos^{47}(a) + 58552888000604040300897239040\, \cos^{45}(a) - 24604142241340407426947481600\, \cos^{43}(a) + 9105549362266552420710481920\, \cos^{41}(a) - 2962916855975624200389918720\, \cos^{39}(a) + 845732430769621545642885120\, \cos^{37}(a) - 211116116526449756146237440\, \cos^{35}(a) + 45911582358639475477708800\, \cos^{33}(a) - 8657612673343443947225088\, \cos^{31}(a) + 1407618843742902599811072\, \cos^{29}(a) - 195985339698085890097152\, \cos^{27}(a) + 23177511534376060452864\, \cos^{25}(a) - 2305455391350403891200\, \cos^{23}(a) + 190614449023415746560\, \cos^{21}(a) - 12912591708037840896\, \cos^{19}(a) + 704098591222726656\, \cos^{17}(a) - 30226454673702912\, \cos^{15}(a) + 993046852546560\, \cos^{13}(a) - 24055172204544\, \cos^{11}(a) + 408343972608\, \cos^{9}(a) - 4514859648\, \cos^{7}(a) + 29012256\, \cos^{5}(a) - 88560\, \cos^{3}(a) + 81\, \cos(a)$ $\cos(82a) = 2417851639229258349412352\, \cos^{82}(a) - 49565958604199796162953216\, \cos^{80}(a) + 489463841216472987109163008\, \cos^{78}(a) - 3100970285175249747444760576\, \cos^{76}(a) + 14163085437098496442656358400\, \cos^{74}(a) - 49681160734549401157941329920\, \cos^{72}(a) + 139237989953408190087388200960\, \cos^{70}(a) - 320247376892838837200992862208\, \cos^{68}(a) + 616151625474566614141775118336\, \cos^{66}(a) - 1005818292726746870117281300480\, \cos^{64}(a) + 1408145609817445618164193820672\, \cos^{62}(a) - 1704739659516510668340903018496\, \cos^{60}(a) + 1796064998419180882716308537344\, \cos^{58}(a) - 1654902699212322184576398000128\, \cos^{56}(a) + 1338524242009966472819145441280\, \cos^{54}(a) - 952949348416050757514525933568\, \cos^{52}(a) + 598300585227122776734972248064\, \cos^{50}(a) - 331637202218654027828208599040\, \cos^{48}(a) + 162364046919549367790893793280\, \cos^{46}(a) - 70194982690782621413356339200\, \cos^{44}(a) + 26775989365113048329449635840\, \cos^{42}(a) - 8998488229259303127110123520\, \cos^{40}(a) + 2658644249553885014827991040\, \cos^{38}(a) - 688661351303014430885806080\, \cos^{36}(a) + 155839314411242489747865600\, \cos^{34}(a) - 30675738731476153245106176\, \cos^{32}(a) + 5224988465251432695595008\, \cos^{30}(a) - 765276088344906808950784\, \cos^{28}(a) + 95659511043113351118848\, \cos^{26}(a) - 10113643815553623654400\, \cos^{24}(a) + 894668491375897477120\, \cos^{22}(a) - 65360032102413762560\, \cos^{20}(a) + 3880751906080817152\, \cos^{18}(a) - 183597724684713984\, \cos^{16}(a) + 6749916348702720\, \cos^{14}(a) - 186699813900288\, \cos^{12}(a) + 3720467305984\, \cos^{10}(a) - 50276585216\, \cos^{8}(a) + 420976192\, \cos^{6}(a) - 1882720\, \cos^{4}(a) + 3362\, \cos^{2}(a) - 1$ $\cos(83a) = 4835703278458516698824704\, \cos^{83}(a) - 100340843028014221500612608\, \cos^{81}(a) + 1003408430280142215006126080\, \cos^{79}(a) - 6440627861860662842570571776\, \cos^{77}(a) + 29818476461715663730002362368\, \cos^{75}(a) - 106084964334949957500969943040\, \cos^{73}(a) + 301722171290312216788472954880\, \cos^{71}(a) - 704680071105663401052025651200\, \cos^{69}(a) + 1377649539011571949056710148096\, \cos^{67}(a) - 2286774122188127311835124727808\, \cos^{65}(a) + 3257869982295414252477437968384\, \cos^{63}(a) - 4016805603171050527202295676928\, \cos^{61}(a) + 4313822918898487361960211906560\, \cos^{59}(a) - 4055467590239182349623034380288\, \cos^{57}(a) + 3350168878893237593166854488064\, \cos^{55}(a) - 2438725875076695012672930840576\, \cos^{53}(a) + 1567427358141644462436240064512\, \cos^{51}(a) - 890583726216843444566045491200\, \cos^{49}(a) + 447575411124362346499858759680\, \cos^{47}(a) - 198942853382169283127609917440\, \cos^{45}(a) + 78156120971566504085846753280\, \cos^{43}(a) - 27102525820785158674930728960\, \cos^{41}(a) + 8280205355083394230045900800\, \cos^{39}(a) - 2223055133375650407414497280\, \cos^{37}(a) + 522794745348934735641968640\, \cos^{35}(a) - 107263059821591781967921152\, \cos^{33}(a) + 19107589603846309338415104\, \cos^{31}(a) - 2938171020432716217712640\, \cos^{29}(a) + 387304361784312592334848\, \cos^{27}(a) - 43404799165483307761664\, \cos^{25}(a) + 4094792374102198845440\, \cos^{23}(a) - 321334513228243271680\, \cos^{21}(a) + 20674095520199475200\, \cos^{19}(a) - 1071294040592154624\, \cos^{17}(a) + 43726287371108352\, \cos^{15}(a) - 1366446480347136\, \cos^{13}(a) + 31496106816512\, \cos^{11}(a) - 508897143040\, \cos^{9}(a) + 5356812032\, \cos^{7}(a) - 32777696\, \cos^{5}(a) + 95284\, \cos^{3}(a) - 83\, \cos(a)$ $\cos(84a) = 9671406556917033397649408\, \cos^{84}(a) - 203099537695257701350637568\, \cos^{82}(a) + 2056382819164484226175205376\, \cos^{80}(a) - 13370719564937798672250306560\, \cos^{78}(a) + 62737923208606577207449485312\, \cos^{76}(a) - 226333014106998411444596244480\, \cos^{74}(a) + 653125503315173834734887239680\, \cos^{72}(a) - 1548598132164734992191439503360\, \cos^{70}(a) + 3075546454915982735314413158400\, \cos^{68}(a) - 5189699869850821237812024573952\, \cos^{66}(a) + 7521558257317575375072157237248\, \cos^{64}(a) - 9441756816159546672568785174528\, \cos^{62}(a) + 10332385497313485392261326831616\, \cos^{60}(a) - 9907000178897545581962377297920\, \cos^{58}(a) + 8355240456998797370910106976256\, \cos^{56}(a) - 6215975992163356498165007122432\, \cos^{54}(a) + 4087804064699339682387006062592\, \cos^{52}(a) - 2379468037660809665867063230464\, \cos^{50}(a) + 1226788024467378720827926118400\, \cos^{48}(a) - 560249753683887934046113628160\, \cos^{46}(a) + 226507224633915629585049845760\, \cos^{44}(a) - 80981041006683365679311093760\, \cos^{42}(a) + 25558898939426091587201925120\, \cos^{40}(a) - 7104754516305185829656985600\, \cos^{38}(a) + 1734250842000883902169743360\, \cos^{36}(a) - 370365434054426053683707904\, \cos^{34}(a) + 68890917939168771921936384\, \cos^{32}(a) - 11101330506116865131020288\, \cos^{30}(a) + 1539884811913531993620480\, \cos^{28}(a) - 182469109374079966642176\, \cos^{26}(a) + 18303228563758021345280\, \cos^{24}(a) - 1537337517832384020480\, \cos^{22}(a) + 106708223142812712960\, \cos^{20}(a) - 6023339987265126400\, \cos^{18}(a) + 271050299426930688\, \cos^{16}(a) - 9482809309396992\, \cos^{14}(a) + 249692027533312\, \cos^{12}(a) - 4738261592064\, \cos^{10}(a) + 60990209280\, \cos^{8}(a) - 486531584\, \cos^{6}(a) + 2073288\, \cos^{4}(a) - 3528\, \cos^{2}(a) + 1$ $\cos(85a) = 19342813113834066795298816\, \cos^{85}(a) - 411034778668973919400099840\, \cos^{83}(a) + 4213106481356982673851023360\, \cos^{81}(a) - 27744847560155739559506739200\, \cos^{79}(a) + 131916474279073817257469542400\, \cos^{77}(a) - 482484504675712486619194851328\, \cos^{75}(a) + 1412335970965297626970744422400\, \cos^{73}(a) - 3398918435619782201171351961600\, \cos^{71}(a) + 6855772980937628871680851968000\, \cos^{69}(a) - 11757049278713214424680759296000\, \cos^{67}(a) + 17329890636823278061979439202304\, \cos^{65}(a) - 22141383614614507597615008317440\, \cos^{63}(a) + 24681576597798021311724949340160\, \cos^{61}(a) - 24127823276693578525884966502400\, \cos^{59}(a) + 20765948504236777091443248332800\, \cos^{57}(a) - 15782120863219950589496868732928\, \cos^{55}(a) + 10614334004475374377446942965760\, \cos^{53}(a) - 6326363433463263794170366525440\, \cos^{51}(a) + 3344159775151600886221897728000\, \cos^{49}(a) - 1568074918492138214592086016000\, \cos^{47}(a) + 651957302650000542297709608960\, \cos^{45}(a) - 240118202984933235444468940800\, \cos^{43}(a) + 78220323699637341849334579200\, \cos^{41}(a) - 22489714387693765889359872000\, \cos^{39}(a) + 5691556817377418211753984000\, \cos^{37}(a) - 1263525613457786843009384448\, \cos^{35}(a) + 245044895699929325811793920\, \cos^{33}(a) - 41310250616080039600455680\, \cos^{31}(a) + 6017940644259780204953600\, \cos^{29}(a) - 752242580532472525619200\, \cos^{27}(a) + 80011256292999350452224\, \cos^{25}(a) - 7169467409766966886400\, \cos^{23}(a) + 534750959513868697600\, \cos^{21}(a) - 32720775494729728000\, \cos^{19}(a) + 1613394639446016000\, \cos^{17}(a) - 62691905989902336\, \cos^{15}(a) + 1865830535413760\, \cos^{13}(a) - 40972630000640\, \cos^{11}(a) + 630877561600\, \cos^{9}(a) - 6329875200\, \cos^{7}(a) + 36924272\, \cos^{5}(a) - 102340\, \cos^{3}(a) + 85\, \cos(a)$ $\cos(86a) = 38685626227668133590597632\, \cos^{86}(a) - 831740963894864872197849088\, \cos^{84}(a) + 8629312500409223049052684288\, \cos^{82}(a) - 57546077939475963345188683776\, \cos^{80}(a) + 277203668123085433187189391360\, \cos^{78}(a) - 1027706932560031550445839187968\, \cos^{76}(a) + 3051004956037593665386085089280\, \cos^{74}(a) - 7450962374554738237077591162880\, \cos^{72}(a) + 15260144094039992735553143439360\, \cos^{70}(a) - 26589645012342411584675931750400\, \cos^{68}(a) + 39849481143497377361770902978560\, \cos^{66}(a) - 51804325486546590570302173872128\, \cos^{64}(a) + 58804910011755589296018683854848\, \cos^{62}(a) - 58588032050700642444031259836416\, \cos^{60}(a) + 51438897187371099764848873963520\, \cos^{58}(a) - 39919482183438698549903844442112\, \cos^{56}(a) + 27444644001114105253058893053952\, \cos^{54}(a) - 16740530931625867270727739113472\, \cos^{52}(a) + 9067787587964011438310858686464\, \cos^{50}(a) - 4362937861451655150012098150400\, \cos^{48}(a) + 1864164358983889018641532846080\, \cos^{46}(a) - 706743630603782100473987727360\, \cos^{44}(a) + 237421688405958049377980252160\, \cos^{42}(a) - 70538327714813623365921669120\, \cos^{40}(a) + 18487868151060022253164953600\, \cos^{38}(a) - 4261302068916457588188512256\, \cos^{36}(a) + 860455225454284705307295744\, \cos^{34}(a) - 151511419171328851122847744\, \cos^{32}(a) + 23137211794636425540927488\, \cos^{30}(a) - 3044369972978477044858880\, \cos^{28}(a) + 342491621960078667546624\, \cos^{26}(a) - 32642163383291955118080\, \cos^{24}(a) + 2606839436860121415680\, \cos^{22}(a) - 172149774132272168960\, \cos^{20}(a) + 9250129266157158400\, \cos^{18}(a) - 396434111406735360\, \cos^{16}(a) + 13214470380224512\, \cos^{14}(a) - 331637287534592\, \cos^{12}(a) + 6000016715264\, \cos^{10}(a) - 73649959680\, \cos^{8}(a) + 560380128\, \cos^{6}(a) - 2277968\, \cos^{4}(a) + 3698\, \cos^{2}(a) - 1$ $\cos(87a) = 77371252455336267181195264\, \cos^{87}(a) - 1682824740903563811190996992\, \cos^{85}(a) + 17669659779487420017505468416\, \cos^{83}(a) - 119305262360308909364228390912\, \cos^{81}(a) + 582152183806326605933885521920\, \cos^{79}(a) - 2187330339399136918149147918336\, \cos^{77}(a) + 6584494416750899817391365029888\, \cos^{75}(a) - 16314260720074774101125926748160\, \cos^{73}(a) + 33919206623699767672277638840320\, \cos^{71}(a) - 60035063005622452041032715468800\, \cos^{69}(a) + 91456011565707969148222565253120\, \cos^{67}(a) - 120938541609916459202583786946560\, \cos^{65}(a) + 139751203638125686189652376027136\, \cos^{63}(a) - 141857640699199306199787469012992\, \cos^{61}(a) + 127005617651435778055582714429440\, \cos^{59}(a) - 100604912871114174191250937217024\, \cos^{57}(a) + 70671408865448161095614654840832\, \cos^{55}(a) - 44095395867727108918902421192704\, \cos^{53}(a) + 24461938609391286670792083898368\, \cos^{51}(a) - 12070035498054911186246094028800\, \cos^{49}(a) + 5296403636459916251875151708160\, \cos^{47}(a) - 2065444563857564743245685063680\, \cos^{45}(a) + 714961579796849334200429445120\, \cos^{43}(a) - 219296979129264588581177917440\, \cos^{41}(a) + 59465450689813810395689779200\, \cos^{39}(a) - 14214160955210333388131008512\, \cos^{37}(a) + 2984436064366356253623975936\, \cos^{35}(a) - 548067734042587028057489408\, \cos^{33}(a) + 87584674205352890682310656\, \cos^{31}(a) - 12106680590216734294671360\, \cos^{29}(a) + 1437225824452629860712448\, \cos^{27}(a) - 145295583059583260688384\, \cos^{25}(a) + 12383146283487209717760\, \cos^{23}(a) - 879050507778413035520\, \cos^{21}(a) + 51221034027044044800\, \cos^{19}(a) - 2406262862259486720\, \cos^{17}(a) + 89120846750351360\, \cos^{15}(a) - 2529105110482944\, \cos^{13}(a) + 52972663431168\, \cos^{11}(a) - 778177480960\, \cos^{9}(a) + 7450635456\, \cos^{7}(a) - 41480208\, \cos^{5}(a) + 109736\, \cos^{3}(a) - 87\, \cos(a)$ $\cos(88a) = 154742504910672534362390528\, \cos^{88}(a) - 3404335108034795755972591616\, \cos^{86}(a) + 36171060522869704907208785920\, \cos^{84}(a) - 247239837221027041777509466112\, \cos^{82}(a) + 1221850445552129175212959727616\, \cos^{80}(a) - 4651864346921359269485485228032\, \cos^{78}(a) + 14196695766061831185228569247744\, \cos^{76}(a) - 35679526396187141867637938585600\, \cos^{74}(a) + 75289375621954273581632868843520\, \cos^{72}(a) - 135330270105284896817618574376960\, \cos^{70}(a) + 209501668143758349881121062256640\, \cos^{68}(a) - 281726564363330295766938476871680\, \cos^{66}(a) + 331306732762797962949606925926400\, \cos^{64}(a) - 342520191410154201695593621880832\, \cos^{62}(a) + 312599267353572198555196688695296\, \cos^{60}(a) - 252648722929599448147350748397568\, \cos^{58}(a) + 181262299914335020741133154123776\, \cos^{56}(a) - 115635435736568323090863735439360\, \cos^{54}(a) + 65664408150408440612311906910208\, \cos^{52}(a) - 33207858584073833810803046744064\, \cos^{50}(a) + 14955745134371487653762401566720\, \cos^{48}(a) - 5995053486699018505132902973440\, \cos^{46}(a) + 2136666790197480768874846617600\, \cos^{44}(a) - 676015646664487226540336087040\, \cos^{42}(a) + 189469229094441244157301227520\, \cos^{40}(a) - 46916190061480689029426970624\, \cos^{38}(a) + 10230174197649170095436464128\, \cos^{36}(a) - 1956590693539458761422274560\, \cos^{34}(a) + 326680767582034632487469056\, \cos^{32}(a) - 47350572975069894130270208\, \cos^{30}(a) + 5918821621883736766283776\, \cos^{28}(a) - 633082788079245188923392\, \cos^{26}(a) + 57408455950266374553600\, \cos^{24}(a) - 4364940452416947486720\, \cos^{22}(a) + 274591842186360258560\, \cos^{20}(a) - 14062654990676131840\, \cos^{18}(a) + 574675804907438080\, \cos^{16}(a) - 18272680601190400\, \cos^{14}(a) + 437582614396928\, \cos^{12}(a) - 7556371677184\, \cos^{10}(a) + 88551230592\, \cos^{8}(a) - 643340544\, \cos^{6}(a) + 2497440\, \cos^{4}(a) - 3872\, \cos^{2}(a) + 1$ $\cos(89a) = 309485009821345068724781056\, \cos^{89}(a) - 6886041468524927779126378496\, \cos^{87}(a) + 74024945786642973625608568832\, \cos^{85}(a) - 512149334221541503572524400640\, \cos^{83}(a) + 2563006153464567259790147846144\, \cos^{81}(a) - 9885880877649045144904855977984\, \cos^{79}(a) + 30580721871522799288606286413824\, \cos^{77}(a) - 77943547209125183552667242201088\, \cos^{75}(a) + 166893011963983321264391664435200\, \cos^{73}(a) - 304579746834269561307514787594240\, \cos^{71}(a) + 479038399293139151803274839982080\, \cos^{69}(a) - 654909140292368560682099518996480\, \cos^{67}(a) + 783552007135512385101797638799360\, \cos^{65}(a) - 824791586458434089580839619788800\, \cos^{63}(a) + 767056175406343703310180846403584\, \cos^{61}(a) - 632303063510634674350284211224576\, \cos^{59}(a) + 463129512699784215673517245464576\, \cos^{57}(a) - 301942280338584807277342125719552\, \cos^{55}(a) + 175424212168543990143526235013120\, \cos^{53}(a) - 90877655777538954292398177386496\, \cos^{51}(a) + 41981525766797886493770897162240\, \cos^{49}(a) - 17286510609857953262140957655040\, \cos^{47}(a) + 6338778144252526280995378298880\, \cos^{45}(a) - 2066992873125823787281101619200\, \cos^{43}(a) + 598235437318147076895780372480\, \cos^{41}(a) - 153297830812775188454543720448\, \cos^{39}(a) + 34674509350508673579003936768\, \cos^{37}(a) - 6897617451445273776468525056\, \cos^{35}(a) + 1201429269206656293032427520\, \cos^{33}(a) - 182285820155492678942851072\, \cos^{31}(a) + 23944323833984207827238912\, \cos^{29}(a) - 2703391400611120238559232\, \cos^{27}(a) + 260112494960116009795584\, \cos^{25}(a) - 21113027188321104691200\, \cos^{23}(a) + 1428234192151133552640\, \cos^{21}(a) - 79346344008396308480\, \cos^{19}(a) + 3555614472074362880\, \cos^{17}(a) - 125666207952732160\, \cos^{15}(a) + 3404270339276800\, \cos^{13}(a) - 68085406785536\, \cos^{11}(a) + 955279942144\, \cos^{9}(a) - 8737316544\, \cos^{7}(a) + 46475088\, \cos^{5}(a) - 117480\, \cos^{3}(a) + 89\, \cos(a)$ $\cos(90a) = 618970019642690137449562112\, \cos^{90}(a) - 13926825441960528092615147520\, \cos^{88}(a) + 151454226681320743007189729280\, \cos^{86}(a) - 1060469728965952712052257587200\, \cos^{84}(a) + 5373252144150161561357805158400\, \cos^{82}(a) - 20993612200850219465022671683584\, \cos^{80}(a) + 65813308089966957846698058055680\, \cos^{78}(a) - 170083790184312198290563053649920\, \cos^{76}(a) + 369465550324153784396421267456000\, \cos^{74}(a) - 684448869290493396196662444032000\, \cos^{72}(a) + 1093407068691563200424168254341120\, \cos^{70}(a) - 1519319948728495471245320100249600\, \cos^{68}(a) + 1848830578634355065970533754470400\, \cos^{66}(a) - 1980889905679666142111286165504000\, \cos^{64}(a) + 1876632542222841608315955314688000\, \cos^{62}(a) - 1577205394374841547255765111144448\, \cos^{60}(a) + 1178907748329167879494385239326720\, \cos^{58}(a) - 785146860591504635295817405562880\, \cos^{56}(a) + 466483860073656303377916205465600\, \cos^{54}(a) - 247419719705486349197108261683200\, \cos^{52}(a) + 117170910117669606798344841068544\, \cos^{50}(a) - 49528766354087394178044316876800\, \cos^{48}(a) + 18672609775204071067123659571200\, \cos^{46}(a) - 6270652536449128343437049856000\, \cos^{44}(a) + 1872486521300781380331896832000\, \cos^{42}(a) - 496064890719991621066388668416\, \cos^{40}(a) + 116265208762498036187434844160\, \cos^{38}(a) - 24025409100539717648373514240\, \cos^{36}(a) + 4359449231952771347487129600\, \cos^{34}(a) - 691252407893019990373171200\, \cos^{32}(a) + 95239220643038309784748032\, \cos^{30}(a) - 11325604423105977243402240\, \cos^{28}(a) + 1153307777999477208514560\, \cos^{26}(a) - 99634510326908583936000\, \cos^{24}(a) + 7221408836719214592000\, \cos^{22}(a) - 433284530203152875520\, \cos^{20}(a) + 21173883934824857600\, \cos^{18}(a) - 826008220812902400\, \cos^{16}(a) + 25081221279744000\, \cos^{14}(a) - 573753427968000\, \cos^{12}(a) + 9466931561472\, \cos^{10}(a) - 106025863680\, \cos^{8}(a) + 736290720\, \cos^{6}(a) - 2732400\, \cos^{4}(a) + 4050\, \cos^{2}(a) - 1$ $\cos(91a) = 1237940039285380274899124224\, \cos^{91}(a) - 28163135893742401253955076096\, \cos^{89}(a) + 309794494831166413793505837056\, \cos^{87}(a) - 2194964403718548397730123743232\, \cos^{85}(a) + 11258653622521864626288134717440\, \cos^{83}(a) - 44550230555165006189835491213312\, \cos^{81}(a) + 141512497057582960838300972089344\, \cos^{79}(a) - 370748302240147195869732393713664\, \cos^{77}(a) + 816874647857432752345509777113088\, \cos^{75}(a) - 1535790750544970113657716552499200\, \cos^{73}(a) + 2491393884217395962155851296276480\, \cos^{71}(a) - 3517678296750130094293915040481280\, \cos^{69}(a) + 4352570297561078692623167027937280\, \cos^{67}(a) - 4745331818494844669324369969807360\, \cos^{65}(a) + 4578056670904117306212750249164800\, \cos^{63}(a) - 3921466964156026797821711068692480\, \cos^{61}(a) + 2990118560168970433339054689878016\, \cos^{59}(a) - 2033423233882793486265152056590336\, \cos^{57}(a) + 1234910000485897414033174536650752\, \cos^{55}(a) - 670263651579516688537742758379520\, \cos^{53}(a) + 325219476012878167889087859523584\, \cos^{51}(a) - 141039058474972674849859530915840\, \cos^{49}(a) + 54631730160266095396388276797440\, \cos^{47}(a) - 18880083217150782967869478010880\, \cos^{45}(a) + 5811965915727386547944895283200\, \cos^{43}(a) - 1590365218758130319028557709312\, \cos^{41}(a) + 385828248337771260829413408768\, \cos^{39}(a) - 82725327551588108875750965248\, \cos^{37}(a) + 15616515915350816471442784256\, \cos^{35}(a) - 2583934084992696273778769920\, \cos^{33}(a) + 372764261441569298512347136\, \cos^{31}(a) - 46595532680196162314043392\, \cos^{29}(a) + 5010006956610074655588352\, \cos^{27}(a) - 459381515613933177667584\, \cos^{25}(a) + 35555844861759533875200\, \cos^{23}(a) - 2294803252557439303680\, \cos^{21}(a) + 121694111878046023680\, \cos^{19}(a) - 5207630913700167680\, \cos^{17}(a) + 175828650512220160\, \cos^{15}(a) - 4551777195212800\, \cos^{13}(a) + 87019269908480\, \cos^{11}(a) - 1167331669504\, \cos^{9}(a) + 10209897984\, \cos^{7}(a) - 51939888\, \cos^{5}(a) + 125580\, \cos^{3}(a) - 91\, \cos(a)$ $\cos(92a) = 2475880078570760549798248448\, \cos^{92}(a) - 56945241807127492645359714304\, \cos^{90}(a) + 633515815104293355679626821632\, \cos^{88}(a) - 4541383034118417538467437215744\, \cos^{86}(a) + 23577776974009681964628527022080\, \cos^{84}(a) - 94473713254480173941028787585024\, \cos^{82}(a) + 304018606316016141141624615862272\, \cos^{80}(a) - 807309912570261349586162845483008\, \cos^{78}(a) + 1803833085899177702981582607876096\, \cos^{76}(a) - 3441047051414094011711854372454400\, \cos^{74}(a) + 5667236637725285320508365036584960\, \cos^{72}(a) - 8128763662191823389011998335303680\, \cos^{70}(a) + 10224460543850652856491654156124160\, \cos^{68}(a) - 11339494215624044404619273694085120\, \cos^{66}(a) + 11137003247487900754536786663833600\, \cos^{64}(a) - 9719566470534895203959377452072960\, \cos^{62}(a) + 7557442514712782413933874490900480\, \cos^{60}(a) - 5245754216094754852024689352507392\, \cos^{58}(a) + 3254966861563299463362166478864384\, \cos^{56}(a) - 1807011163232689680453401722224640\, \cos^{54}(a) + 897858671731242684975283980730368\, \cos^{52}(a) - 399249027067614956498063902900224\, \cos^{50}(a) + 158792226674619584970820870471680\, \cos^{48}(a) - 56432776209505637002862615592960\, \cos^{46}(a) + 17894584367903901439326840422400\, \cos^{44}(a) - 5053216958817042018389012250624\, \cos^{42}(a) + 1267721387395534142725215485952\, \cos^{40}(a) - 281715863865674253938936774656\, \cos^{38}(a) + 55258440931241350591259082752\, \cos^{36}(a) - 9527317401938163895044669440\, \cos^{34}(a) + 1436780930776158587397865472\, \cos^{32}(a) - 188430286003430634412834816\, \cos^{30}(a) + 21345618336326126554578944\, \cos^{28}(a) - 2072070809227343563849728\, \cos^{26}(a) + 170746200050427651686400\, \cos^{24}(a) - 11811015341834093199360\, \cos^{22}(a) + 676672753959244922880\, \cos^{20}(a) - 31589145762225192960\, \cos^{18}(a) + 1177665521837342720\, \cos^{16}(a) - 34184775670169600\, \cos^{14}(a) + 747791967784960\, \cos^{12}(a) - 11801594900480\, \cos^{10}(a) + 126445659648\, \cos^{8}(a) - 840170496\, \cos^{6}(a) + 2983560\, \cos^{4}(a) - 4232\, \cos^{2}(a) + 1$ $\cos(93a) = 4951760157141521099596496896\, \cos^{93}(a) - 115128423653540365565618552832\, \cos^{91}(a) + 1295194766102329112613208719360\, \cos^{89}(a) - 9392560563068001490728380268544\, \cos^{87}(a) + 49350518351737912326987177787392\, \cos^{85}(a) - 200206080131482212508345709887488\, \cos^{83}(a) + 652587443187197288473084722937856\, \cos^{81}(a) - 1756132322198105660010626663055360\, \cos^{79}(a) + 3978414474038502601832897609465856\, \cos^{77}(a) - 7698968750685620775769218522021888\, \cos^{75}(a) + 12870264025995540754674446625669120\, \cos^{73}(a) - 18748921208601042740179847966883840\, \cos^{71}(a) + 23966599384451435807277223352729600\, \cos^{69}(a) - 27031558728809167501861714416107520\, \cos^{67}(a) + 27019338313470646178397943297474560\, \cos^{65}(a) - 24017189611973907714131505153310720\, \cos^{63}(a) + 19036351993581591625689460050493440\, \cos^{61}(a) - 13481626992358480137388433394892800\, \cos^{59}(a) + 8543356957009392412989485014319104\, \cos^{57}(a) - 4848932326951276774939977981100032\, \cos^{55}(a) + 2465980995042002058488310719840256\, \cos^{53}(a) - 1123717530148108080885215665324032\, \cos^{51}(a) + 458623511824211844791501271859200\, \cos^{49}(a) - 167497282579277369402113507983360\, \cos^{47}(a) + 54669251952958585846523158855680\, \cos^{45}(a) - 15918399833361470584722919784448\, \cos^{43}(a) + 4125807993549198604478988681216\, \cos^{41}(a) - 949259976069119768707286958080\, \cos^{39}(a) + 193242209414070810058269130752\, \cos^{37}(a) - 34671150719227144261532123136\, \cos^{35}(a) + 5457495946545013448574500864\, \cos^{33}(a) - 749624833448430567338016768\, \cos^{31}(a) + 89286769352848415423201280\, \cos^{29}(a) - 9154148575064761783287808\, \cos^{27}(a) + 800873915714788481040384\, \cos^{25}(a) - 59177875545427720273920\, \cos^{23}(a) + 3648148760475929149440\, \cos^{21}(a) - 184872403402496409600\, \cos^{19}(a) + 7562961957374853120\, \cos^{17}(a) - 244198201852559360\, \cos^{15}(a) + 6047361130782720\, \cos^{13}(a) - 110622459709440\, \cos^{11}(a) + 1420222988800\, \cos^{9}(a) - 11890238976\, \cos^{7}(a) + 57907008\, \cos^{5}(a) - 134044\, \cos^{3}(a) + 93\, \cos(a)$ $\cos(94a) = 9903520314283042199192993792\, \cos^{94}(a) - 232732727385651491681035354112\, \cos^{92}(a) + 2647334774011785717871777153024\, \cos^{90}(a) - 19418636941240296337136387358720\, \cos^{88}(a) + 103242419737594242192441792790528\, \cos^{86}(a) - 423989937236974106981319946797056\, \cos^{84}(a) + 1399648599628874750887198233460736\, \cos^{82}(a) - 3816283250712227461162877941972992\, \cos^{80}(a) + 8764138860647266553251958064414720\, \cos^{78}(a) - 17201770587270419254520019651919872\, \cos^{76}(a) + 29181575103405175521060747623792640\, \cos^{74}(a) - 43165079054927370800868060970352640\, \cos^{72}(a) + 56061962431094695003566445040762880\, \cos^{70}(a) - 64287578001468987860215082988339200\, \cos^{68}(a) + 65378170842565336761415160289034240\, \cos^{66}(a) - 59171382471435716182799796970455040\, \cos^{64}(a) + 47792270457698078455338297553059840\, \cos^{62}(a) - 34520696499429742688710741280686080\, \cos^{60}(a) + 22332468130113539678003659381145600\, \cos^{58}(a) - 12952831515465853013242122441064448\, \cos^{56}(a) + 6738973153316693797430023161905152\, \cos^{54}(a) - 3145293732027458846745715311378432\, \cos^{52}(a) + 1316496050716038646081066446618624\, \cos^{50}(a) - 493786791833174323775047886438400\, \cos^{48}(a) + 165771280115422808695908933304320\, \cos^{46}(a) - 49731384034626842608772679991296\, \cos^{44}(a) + 13304832945915439227346989613056\, \cos^{42}(a) - 3166241339533773680139789402112\, \cos^{40}(a) + 668200282693815874055475036160\, \cos^{38}(a) - 124600742369695639114323329024\, \cos^{36}(a) + 20442309295028190792193671168\, \cos^{34}(a) - 2936030597673019722073899008\, \cos^{32}(a) + 367003824709127465259237376\, \cos^{30}(a) - 39653915486455650121154560\, \cos^{28}(a) + 3673818640656920525930496\, \cos^{26}(a) - 289101951141283092234240\, \cos^{24}(a) + 19107312862785951498240\, \cos^{22}(a) - 1046417560764237742080\, \cos^{20}(a) + 46715069676974899200\, \cos^{18}(a) - 1666061925542461440\, \cos^{16}(a) + 46279497931735040\, \cos^{14}(a) - 969036887203840\, \cos^{12}(a) + 14642040878080\, \cos^{10}(a) - 150226137600\, \cos^{8}(a) + 955984512\, \cos^{6}(a) - 3251648\, \cos^{4}(a) + 4418\, \cos^{2}(a) - 1$ $\cos(95a) = 19807040628566084398385987584\, \cos^{95}(a) - 470417214928444504461667205120\, \cos^{93}(a) + 5409797971677111801309172858880\, \cos^{91}(a) - 40132468648582921786885983436800\, \cos^{89}(a) + 215877400038256485875611965849600\, \cos^{87}(a) - 897330392825686126289627071381504\, \cos^{85}(a) + 2999503279389231714282742176808960\, \cos^{83}(a) - 8285153944611652210798840606883840\, \cos^{81}(a) + 19284410043492638766514542791884800\, \cos^{79}(a) - 38381955648579341110872936913305600\, \cos^{77}(a) + 66062118957495971817890713769607168\, \cos^{75}(a) - 99200422135850282356410568566374400\, \cos^{73}(a) + 130872846070790432747312738048409600\, \cos^{71}(a) - 152541755387389411527707389329408000\, \cos^{69}(a) + 157787900413939841024692034994176000\, \cos^{67}(a) - 145362103256342078543997537238384640\, \cos^{65}(a) + 119601730527370064624808100259430400\, \cos^{63}(a) - 88077744992441077003110942611865600\, \cos^{61}(a) + 58146563252585559493395752157184000\, \cos^{59}(a) - 34449019987941098439473729896448000\, \cos^{57}(a) + 18326878633584664369800024304910336\, \cos^{55}(a) - 8756568459096919751979741342597120\, \cos^{53}(a) + 3756709631580185373047348558561280\, \cos^{51}(a) - 1446197095490560492341597044736000\, \cos^{49}(a) + 499039842810122986793931374592000\, \cos^{47}(a) - 154132020022212271064068518838272\, \cos^{45}(a) + 42528065725192349039416899010560\, \cos^{43}(a) - 10458290672616745964758567485440\, \cos^{41}(a) + 2285660541456751516818237030400\, \cos^{39}(a) - 442443694153462088286915788800\, \cos^{37}(a) + 75555769309283525845919465472\, \cos^{35}(a) - 11329557141891052892722298880\, \cos^{33}(a) + 1483632482866685497856491520\, \cos^{31}(a) - 168594600325759715665510400\, \cos^{29}(a) + 16501785856378602835148800\, \cos^{27}(a) - 1379077817997354665508864\, \cos^{25}(a) + 97392501270999623270400\, \cos^{23}(a) - 5740983882004404633600\, \cos^{21}(a) + 278302542756446208000\, \cos^{19}(a) - 10895085808459776000\, \cos^{17}(a) + 336757197716029440\, \cos^{15}(a) - 7985434905190400\, \cos^{13}(a) + 139906541465600\, \cos^{11}(a) - 1720675264000\, \cos^{9}(a) + 13802208000\, \cos^{7}(a) - 64410304\, \cos^{5}(a) + 142880\, \cos^{3}(a) - 95\, \cos(a)$ $\cos(96a) = 39614081257132168796771975168\, \cos^{96}(a) - 950737950171172051122527404032\, \cos^{94}(a) + 11052328670739875094299381071872\, \cos^{92}(a) - 82912272071177629291643744026624\, \cos^{90}(a) + 451173437017753268088360319057920\, \cos^{88}(a) - 1897903205388966494771695935553536\, \cos^{86}(a) + 6422996496015437535546804300414976\, \cos^{84}(a) - 17969956488852179172484879447228416\, \cos^{82}(a) + 42385103337697504994191963525742592\, \cos^{80}(a) - 85528050157805948774997831891025920\, \cos^{78}(a) + 149326008502262362890301447191134208\, \cos^{76}(a) - 227582419375105740233881884756541440\, \cos^{74}(a) + 304910771196508236295493537067171840\, \cos^{72}(a) - 361145473205873518058981223699578880\, \cos^{70}(a) + 379863378829348669909599152976691200\, \cos^{68}(a) - 356102377355249493849410234765803520\, \cos^{66}(a) + 298374843526175845432415997489315840\, \cos^{64}(a) - 223947760442580232461560182776791040\, \cos^{62}(a) + 150813823004600861675502245595054080\, \cos^{60}(a) - 91230508105995736556951119174041600\, \cos^{58}(a) + 49606588782635181752842171050885120\, \cos^{56}(a) - 24252110071510533301389505847099392\, \cos^{54}(a) + 10658712995187829592840412428500992\, \cos^{52}(a) - 4208890241697159630764260536090624\, \cos^{50}(a) + 1491866477453420297362910635622400\, \cos^{48}(a) - 474035320159847350824045970980864\, \cos^{46}(a) + 134787515485011540687606478012416\, \cos^{44}(a) - 34221414291148931156864124583936\, \cos^{42}(a) + 7737562422447276713776263462912\, \cos^{40}(a) - 1553087671000740050629306613760\, \cos^{38}(a) + 275712280988262690806162259968\, \cos^{36}(a) - 43101423578810296577638268928\, \cos^{34}(a) + 5903295563406390717786882048\, \cos^{32}(a) - 704193025360646896590258176\, \cos^{30}(a) + 72657487199212855791452160\, \cos^{28}(a) - 6431974276651629856948224\, \cos^{26}(a) + 483886953683282338775040\, \cos^{24}(a) - 30589280626794760765440\, \cos^{22}(a) + 1603022646277130158080\, \cos^{20}(a) - 68505241293894451200\, \cos^{18}(a) + 2339576320974520320\, \cos^{16}(a) - 62250367742115840\, \cos^{14}(a) + 1248849970135040\, \cos^{12}(a) - 18083391406080\, \cos^{10}(a) + 177830553600\, \cos^{8}(a) - 1084805120\, \cos^{6}(a) + 3537408\, \cos^{4}(a) - 4608\, \cos^{2}(a) + 1$ $\cos(97a) = 79228162514264337593543950336\, \cos^{97}(a) - 1921282940970910186643440795648\, \cos^{95}(a) + 22575074556408194693060429348864\, \cos^{93}(a) - 171234342114032370384596660912128\, \cos^{91}(a) + 942479342684089457963606621552640\, \cos^{89}(a) - 4011683810816189475419003836956672\, \cos^{87}(a) + 13743323384856561197383235672211456\, \cos^{85}(a) - 38939416257093590059252501071265792\, \cos^{83}(a) + 93055360620006662199182767658369024\, \cos^{81}(a) - 190340510359104536316510206573936640\, \cos^{79}(a) + 337033972653104066891475831295574016\, \cos^{77}(a) - 521226957707707452285654483282690048\, \cos^{75}(a) + 709021964528866754947397642700718080\, \cos^{73}(a) - 853163792482537468865275185447567360\, \cos^{71}(a) + 912268513046086751346905695282790400\, \cos^{69}(a) - 869992655124438828723512504525783040\, \cos^{67}(a) + 742111790308693769408829532217016320\, \cos^{65}(a) - 567497251412530529547928465813012480\, \cos^{63}(a) + 389705391001642800354115433801973760\, \cos^{61}(a) - 240607579464577032607297990505267200\, \cos^{59}(a) + 133662197553211461945158071998218240\, \cos^{57}(a) - 66831098776605730972579035999109120\, \cos^{55}(a) + 30073994449472578937660566199599104\, \cos^{53}(a) - 12174490114974504634575869630742528\, \cos^{51}(a) + 4429930050397401087067418315980800\, \cos^{49}(a) - 1447110483129817688442023316553728\, \cos^{47}(a) + 423707050992235352439281474863104\, \cos^{45}(a) - 110970894307490211353145148178432\, \cos^{43}(a) + 25933415517511299392311094411264\, \cos^{41}(a) - 5391835883458231618076850257920\, \cos^{39}(a) + 993868256129987469899240308736\, \cos^{37}(a) - 161758616466904119001196003328\, \cos^{35}(a) + 23136148268703834328296062976\, \cos^{33}(a) - 2892018533587979291037007872\, \cos^{31}(a) + 313909574724185427248414720\, \cos^{29}(a) - 29365734409681862549045248\, \cos^{27}(a) + 2346851725363919343058944\, \cos^{25}(a) - 158571062524589144801280\, \cos^{23}(a) + 8947029174558664949760\, \cos^{21}(a) - 415313025344235110400\, \cos^{19}(a) + 15574238450408816640\, \cos^{17}(a) - 461257933200261120\, \cos^{15}(a) + 10483134845460480\, \cos^{13}(a) - 176073324277760\, \cos^{11}(a) + 2076336371200\, \cos^{9}(a) - 15971818240\, \cos^{7}(a) + 71485120\, \cos^{5}(a) - 152096\, \cos^{3}(a) + 97\, \cos(a)$ $\cos(98a) = 158456325028528675187087900672\, \cos^{98}(a) - 3882179963198952542083653566464\, \cos^{96}(a) + 46100887062987561437243386101760\, \cos^{94}(a) - 353521012898804615863492702896128\, \cos^{92}(a) + 1967870957439356545218856987131904\, \cos^{90}(a) - 8474541058650132218926367992971264\, \cos^{88}(a) + 29384549975102088889538167279976448\, \cos^{86}(a) - 84301829010202617654051806442946560\, \cos^{84}(a) + 204080677728865503570850414763966464\, \cos^{82}(a) - 423066124055906577627212376673615872\, \cos^{80}(a) + 759595995464014082557949494482173952\, \cos^{78}(a) - 1191779923917677267461610413756514304\, \cos^{76}(a) + 1645626348432839250128677170157977600\, \cos^{74}(a) - 2011238356161583174026043907962306560\, \cos^{72}(a) + 2185682499298047020752792614265159680\, \cos^{70}(a) - 2119848689078226327356624162028257280\, \cos^{68}(a) + 1840325957972637032667069299199836160\, \cos^{66}(a) - 1433369346351236904528272929115340800\, \cos^{64}(a) + 1003358542445865833169791050380738560\, \cos^{62}(a) - 632028981933754926890098226605588480\, \cos^{60}(a) + 358554903212418660447267263170478080\, \cos^{58}(a) - 183268786335846643698000243049103360\, \cos^{56}(a) + 84400098970455691176710638246297600\, \cos^{54}(a) - 35007693225136838861992151689986048\, \cos^{52}(a) + 13068750342491961804899097168052224\, \cos^{50}(a) - 4386087443713055674246957268729856\, \cos^{48}(a) + 1321449422144318055702608920707072\, \cos^{46}(a) - 356729304099991963393896774369280\, \cos^{44}(a) + 86088245326171529941486313406464\, \cos^{42}(a) - 18521234189363739949929963978752\, \cos^{40}(a) + 3540824183260714990427787231232\, \cos^{38}(a) - 599229513922070928808554266624\, \cos^{36}(a) + 89373720116217965234230394880\, \cos^{34}(a) - 11687332630582349299860897792\, \cos^{32}(a) + 1332012174809017751087087616\, \cos^{30}(a) - 131388956018576580889542656\, \cos^{28}(a) + 11125677727379468543066112\, \cos^{26}(a) - 801029078732460628377600\, \cos^{24}(a) + 48483338975912090664960\, \cos^{22}(a) - 2433648696965600378880\, \cos^{20}(a) + 99653718194712084480\, \cos^{18}(a) - 3262092187375042560\, \cos^{16}(a) + 83216637433036800\, \cos^{14}(a) - 1600996618690560\, \cos^{12}(a) + 22236064148480\, \cos^{10}(a) - 209774190080\, \cos^{8}(a) + 1227775360\, \cos^{6}(a) - 3841600\, \cos^{4}(a) + 4802\, \cos^{2}(a) - 1$ $\cos(99a) = 316912650057057350374175801344\, \cos^{99}(a) - 7843588088912169421760851083264\, \cos^{97}(a) + 94123057066946033061130212999168\, \cos^{95}(a) - 729617100354017426420045835141120\, \cos^{93}(a) + 4106976256992745460822310635175936\, \cos^{91}(a) - 17891561459984353895816342607495168\, \cos^{89}(a) + 62780783761020367254495338396909568\, \cos^{87}(a) - 182346981405261796505486848558104576\, \cos^{85}(a) + 447100771714824597200953330599198720\, \cos^{83}(a) - 939187608731819817453607521005600768\, \cos^{81}(a) + 1709532501287132701432409195538284544\, \cos^{79}(a) - 2720593820488458601814696658808602624\, \cos^{77}(a) + 3812479654573385952543008823598645248\, \cos^{75}(a) - 4731498676852033102999485458625331200\, \cos^{73}(a) + 5224528791078631510370860413977886720\, \cos^{71}(a) - 5151965891202539406060154019339304960\, \cos^{69}(a) + 4550644571069712894057651102925455360\, \cos^{67}(a) - 3608850483011167578465375390447697920\, \cos^{65}(a) + 2574214336304262195887510566574489600\, \cos^{63}(a) - 1653763354869152654134311887013150720\, \cos^{61}(a) + 957717385889414353501832516846223360\, \cos^{59}(a) - 500199770224904749341158558096424960\, \cos^{57}(a) + 235631296717517113326000312491704320\, \cos^{55}(a) - 100089380899746256661644869579571200\, \cos^{53}(a) + 38311990799958428244374063966846976\, \cos^{51}(a) - 13202104937823512435561332853440512\, \cos^{49}(a) + 4090009327418453799847241157967872\, \cos^{47}(a) - 1137165659192219279227075023601664\, \cos^{45}(a) + 283147384959833271236117774991360\, \cos^{43}(a) - 62975883896238779292171022368768\, \cos^{41}(a) + 12473484249979661598932424720384\, \cos^{39}(a) - 2192327283974129327516348841984\, \cos^{37}(a) + 340506056699340049469656793088\, \cos^{35}(a) - 46510813529868532928017858560\, \cos^{33}(a) + 5556042883206014793211183104\, \cos^{31}(a) - 576687486761338589027500032\, \cos^{29}(a) + 51617089864440799635177472\, \cos^{27}(a) - 3948909882828840599814144\, \cos^{25}(a) + 255537740476413326131200\, \cos^{23}(a) - 13814326568489865707520\, \cos^{21}(a) + 614620461733659279360\, \cos^{19}(a) - 22098422825158901760\, \cos^{17}(a) + 627691208066334720\, \cos^{15}(a) - 13685128082841600\, \cos^{13}(a) + 220545452574720\, \cos^{11}(a) - 2495884751360\, \cos^{9}(a) + 18427368960\, \cos^{7}(a) - 79168320\, \cos^{5}(a) + 161700\, \cos^{3}(a) - 99\, \cos(a)$ $\cos(100a) = 633825300114114700748351602688\, \cos^{100}(a) - 15845632502852867518708790067200\, \cos^{98}(a) + 192128294097091018664344079564800\, \cos^{96}(a) - 1505335087771022414277335056384000\, \cos^{94}(a) + 8567473526884295537508113973248000\, \cos^{92}(a) - 37750993877408064336851542202122240\, \cos^{90}(a) + 134036108580690866727917044786790400\, \cos^{88}(a) - 394078512785625681900511864396185600\, \cos^{86}(a) + 978503372439851812055958467641344000\, \cos^{84}(a) - 2082455895192505138478065456775168000\, \cos^{82}(a) + 3842131126630171980492030767750184960\, \cos^{80}(a) - 6200783636440931286187342812099379200\, \cos^{78}(a) + 8816739233064449172547628060953804800\, \cos^{76}(a) - 11108623702136905456127648087408640000\, \cos^{74}(a) + 12460295938318846194767764735918080000\, \cos^{72}(a) - 12489614281703125832873100652943769600\, \cos^{70}(a) + 11221137831217652115471926367879168000\, \cos^{68}(a) - 9058026923994972189597820080095232000\, \cos^{66}(a) + 6581798018959761296303294062264320000\, \cos^{64}(a) - 4310885252184171141438414824407040000\, \cos^{62}(a) + 2547463753712583633893763260298035200\, \cos^{60}(a) - 1358954443662228159129584379363328000\, \cos^{58}(a) + 654531379770880870350000868032512000\, \cos^{56}(a) - 284578860769948204500000377405440000\, \cos^{54}(a) + 111631674825053695350740279623680000\, \cos^{52}(a) - 39472960218138986676021762874933248\, \cos^{50}(a) + 12566106098549963273941439584665600\, \cos^{48}(a) - 3595780740528756614156758967910400\, \cos^{46}(a) + 923024074019658505866132324352000\, \cos^{44}(a) - 212040013118649088525828358144000\, \cos^{42}(a) + 43468202689323063147794813419520\, \cos^{40}(a) - 7925478751208973645460484915200\, \cos^{38}(a) + 1280241627320751027747867852800\, \cos^{36}(a) - 182395347175955031090266112000\, \cos^{34}(a) + 22799418396994378886283264000\, \cos^{32}(a) - 2485387148331694929142087680\, \cos^{30}(a) + 234623135747458180159897600\, \cos^{28}(a) - 19023497493037149742694400\, \cos^{26}(a) + 1312104559685287280640000\, \cos^{24}(a) - 76111992112891822080000\, \cos^{22}(a) + 3662889620432918937600\, \cos^{20}(a) - 143850563845029888000\, \cos^{18}(a) + 4517474603507712000\, \cos^{16}(a) - 110586893598720000\, \cos^{14}(a) + 2042087523840000\, \cos^{12}(a) - 27227833651200\, \cos^{10}(a) + 246628928000\, \cos^{8}(a) - 1386112000\, \cos^{6}(a) + 4165000\, \cos^{4}(a) - 5000\, \cos^{2}(a) + 1$