$\sin (2x)=2\,\cos x\,\sin x$
$\cos (2x)=\cos ^2x-\sin ^2x$
$\sin (3x)=3\,\cos ^2x\,\sin x-\sin ^3x$
$\cos (3x)=\cos ^3x-3\,\cos x\,\sin ^2x$
$\sin (4x)=4\,\cos ^3x\,\sin x-4\,\cos x\,\sin ^3x$
$\cos (4x)=\sin ^4x-6\,\cos ^2x\,\sin ^2x+\cos ^4x$
$\sin (5x)=\sin ^5x-10\,\cos ^2x\,\sin ^3x+5\,\cos ^4x\,\sin x$
$\cos (5x)=5\,\cos x\,\sin ^4x-10\,\cos ^3x\,\sin ^2x+\cos ^5x$
$\sin (6x)=6\,\cos x\,\sin ^5x-20\,\cos ^3x\,\sin ^3x+6\,\cos ^5x\,\sin x$
$\cos (6x)=-\sin ^6x+15\,\cos ^2x\,\sin ^4x-15\,\cos ^4x\,\sin ^2x+\cos ^6x$
$\sin (7x)=-\sin ^7x+21\,\cos ^2x\,\sin ^5x-35\,\cos ^4x\,\sin ^3x+7\,\cos ^6x\,\sin x$
$\cos (7x)=-7\,\cos x\,\sin ^6x+35\,\cos ^3x\,\sin ^4x-21\,\cos ^5x\,\sin ^2x+\cos ^7x$
$\sin (8x)=-8\,\cos x\,\sin ^7x+56\,\cos ^3x\,\sin ^5x-56\,\cos ^5x\,\sin ^3x+8\,\cos ^7x\,\sin x$
$\cos (8x)=\sin ^8x-28\,\cos ^2x\,\sin ^6x+70\,\cos ^4x\,\sin ^4x-28\,\cos ^6x\,\sin ^2x+\cos ^8x$
$\sin (9x)=\sin ^9x-36\,\cos ^2x\,\sin ^7x+126\,\cos ^4x\,\sin ^5x-84\,\cos ^6x\,\sin ^3x+9\,\cos ^8x\,\sin x$
$\cos (9x)=9\,\cos x\,\sin ^8x-84\,\cos ^3x\,\sin ^6x+126\,\cos ^5x\,\sin ^4x-36\,\cos ^7x\,\sin ^2x+\cos ^9x$
${\small \sin (10x)=10\,\cos x\,\sin ^9x-120\,\cos ^3x\,\sin ^7x+252\,\cos ^5x\,\sin ^5x-120\,\cos ^7x\,\sin ^3x+10\,\cos ^9x\,\sin x}$
${\small \cos (10x)=-\sin ^{10}x+45\,\cos ^2x\,\sin ^8x-210\,\cos ^4x\,\sin ^6x+210\,\cos ^6x\,\sin ^4x-45\,\cos ^8x\,\sin ^2x+\cos ^{10}x}$
${\scriptsize \sin (11x)=-\sin ^{11}x+55\,\cos ^2x\,\sin ^9x-330\,\cos ^4x\,\sin ^7x+462\,\cos ^6x\,\sin ^5x-165\,\cos ^8x\,\sin ^3x+11\,\cos ^{10}x\,\sin x}$
${\scriptsize \cos (11x)=-11\,\cos x\,\sin ^{10}x+165\,\cos ^3x\,\sin ^8x-462\,\cos ^5x\,\sin ^6x+330\,\cos ^7x\,\sin ^4x-55\,\cos ^9x\,\sin ^2x+\cos ^{11}x}$
${\tiny \sin (12x)=-12\,\cos x\,\sin ^{11}x+220\,\cos ^3x\,\sin ^9x-792\,\cos ^5x\,\sin ^7x+792\,\cos ^7x\,\sin ^5x-220\,\cos ^9x\,\sin ^3x+12\,\cos ^{11}x\,\sin x}$
${\tiny \cos (12x)=\sin ^{12}x-66\,\cos ^2x\,\sin ^{10}x+495\,\cos ^4x\,\sin ^8x-924\,\cos ^6x\,\sin ^6x+495\,\cos ^8x\,\sin ^4x-66\,\cos ^{10}x\,\sin ^2x+\cos ^{12}x}$
${\tiny \sin (13x)=\sin ^{13}x-78\,\cos ^2x\,\sin ^{11}x+715\,\cos ^4x\,\sin ^9x-1716\,\cos ^6x\,\sin ^7x+1287\,\cos ^8x\,\sin ^5x-286\,\cos ^{10}x\,\sin ^3x+13\,\cos ^{12}x\,\sin x}$
${\tiny \cos (13x)=13\,\cos x\,\sin ^{12}x-286\,\cos ^3x\,\sin ^{10}x+1287\,\cos ^5x\,\sin ^8x-1716\,\cos ^7x\,\sin ^6x+715\,\cos ^9x\,\sin ^4x-78\,\cos ^{11}x\,\sin ^2x+\cos ^{13}x}$
${\tiny \sin (14x)=14\,\cos x\,\sin ^{13}x-364\,\cos ^3x\,\sin ^{11}x+2002\,\cos ^5x\,\sin ^9x-3432\,\cos ^7x\,\sin ^7x+2002\,\cos ^9x\,\sin ^5x-364\,\cos ^{11}x\,\sin ^3x+14\,\cos ^{13}x\,\sin x}$
${\tiny \cos (14x)=-\sin ^{14}x+91\,\cos ^2x\,\sin ^{12}x-1001\,\cos ^4x\,\sin ^{10}x+3003\,\cos ^6x\,\sin ^8x-3003\,\cos ^8x\,\sin ^6x+1001\,\cos ^{10}x\,\sin ^4x-91\,\cos ^{12}x\,\sin ^2x+\cos ^{14}x}$
${\tiny \sin (15x)=-\sin ^{15}x+105\,\cos ^2x\,\sin ^{13}x-1365\,\cos ^4x\,\sin ^{11}x+5005\,\cos ^6x\,\sin ^9x-6435\,\cos ^8x\,\sin ^7x+3003\,\cos ^{10}x\,\sin ^5x-455\,\cos ^{12}x\,\sin ^3x+15\,\cos ^{14}x\,\sin x}$
${\tiny \cos (15x)=-15\,\cos x\,\sin ^{14}x+455\,\cos ^3x\,\sin ^{12}x-3003\,\cos ^5x\,\sin ^{10}x+6435\,\cos ^7x\,\sin ^8x-5005\,\cos ^9x\,\sin ^6x+1365\,\cos ^{11}x\,\sin ^4x-105\,\cos ^{13}x\,\sin ^2x+\cos ^{15}x}$
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