Sous réserve d'erreur(s), on a :
Sachant que :
{ cos x = e i x + e − i x 2 cos 2 x = e 2 i x + 2 + e − 2 i x 4 = cos ( 2 x ) + 1 2 cos 3 x = e 3 i x + 3 e i x + 3 e − i x + e − 3 i x 8 = cos ( 3 x ) + 3 cos ( x ) 4 cos 4 x = e 4 i x + 4 e 2 i x + 6 + 4 e − 2 i x + e − 4 i x 16 = cos ( 4 x ) + 4 cos ( 2 x ) + 3 8 {\displaystyle {\begin{cases}\cos x&={\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}\\\cos ^{2}x&={\frac {\mathrm {e} ^{2\mathrm {i} x}+2+\mathrm {e} ^{-2\mathrm {i} x}}{4}}={\frac {\cos \left(2x\right)+1}{2}}\\\cos ^{3}x&={\frac {\mathrm {e} ^{3\mathrm {i} x}+3\mathrm {e} ^{\mathrm {i} x}+3\mathrm {e} ^{-\mathrm {i} x}+\mathrm {e} ^{-3\mathrm {i} x}}{8}}={\frac {\cos \left(3x\right)+3\cos \left(x\right)}{4}}\\\cos ^{4}x&={\frac {\mathrm {e} ^{4\mathrm {i} x}+4\mathrm {e} ^{2\mathrm {i} x}+6+4\mathrm {e} ^{-2\mathrm {i} x}+\mathrm {e} ^{-4\mathrm {i} x}}{16}}={\frac {\cos \left(4x\right)+4\cos \left(2x\right)+3}{8}}\end{cases}}}
et :
{ sin x = e i x − e − i x 2 i sin 2 x = e 2 i x − 2 + e − 2 i x − 4 = − cos ( 2 x ) + 1 2 sin 3 x = e 3 i x − 3 e i x + 3 e − i x − e − 3 i x − 8 i = − sin ( 3 x ) + 3 sin ( x ) 4 sin 4 x = e 4 i x − 4 e 2 i x + 6 − 4 e − 2 i x + e − 4 i x 16 = cos ( 4 x ) − 4 cos ( 2 x ) + 3 8 {\displaystyle {\begin{cases}\sin x&={\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}\\\sin ^{2}x&={\frac {\mathrm {e} ^{2\mathrm {i} x}-2+\mathrm {e} ^{-2\mathrm {i} x}}{-4}}={\frac {-\cos \left(2x\right)+1}{2}}\\\sin ^{3}x&={\frac {\mathrm {e} ^{3\mathrm {i} x}-3\mathrm {e} ^{\mathrm {i} x}+3\mathrm {e} ^{-\mathrm {i} x}-\mathrm {e} ^{-3\mathrm {i} x}}{-8\mathrm {i} }}={\frac {-\sin \left(3x\right)+3\sin \left(x\right)}{4}}\\\sin ^{4}x&={\frac {\mathrm {e} ^{4\mathrm {i} x}-4\mathrm {e} ^{2\mathrm {i} x}+6-4\mathrm {e} ^{-2\mathrm {i} x}+\mathrm {e} ^{-4\mathrm {i} x}}{16}}={\frac {\cos \left(4x\right)-4\cos \left(2x\right)+3}{8}}\end{cases}}}
On obtient :
{ cos 1 x = [ cos x ] cos 2 x = 1 2 [ cos ( 2 x ) + 1 ] cos 3 x = 1 4 [ cos ( 3 x ) + 3 cos ( x ) ] cos 4 x = 1 8 [ cos ( 4 x ) + 4 cos ( 2 x ) + 3 ] cos 5 x = 1 16 [ cos ( 5 x ) + 5 cos ( 3 x ) + 10 cos ( x ) ] cos 6 x = 1 32 [ cos ( 6 x ) + 6 cos ( 4 x ) + 15 cos ( 2 x ) + 10 ] cos 7 x = 1 64 [ cos ( 7 x ) + 7 cos ( 5 x ) + 21 cos ( 3 x ) + 35 cos ( x ) ] cos 8 x = 1 128 [ cos ( 8 x ) + 8 cos ( 6 x ) + 28 cos ( 4 x ) + 56 cos ( 2 x ) + 35 ] cos 9 x = 1 256 [ cos ( 9 x ) + 9 cos ( 7 x ) + 36 cos ( 5 x ) + 84 cos ( 3 x ) + 126 cos ( x ) ] cos 10 x = 1 512 [ cos ( 10 x ) + 10 cos ( 8 x ) + 45 cos ( 6 x ) + 120 cos ( 4 x ) + 210 cos ( 2 x ) + 126 ] cos 11 x = 1 1024 [ cos ( 11 x ) + 11 cos ( 9 x ) + 55 cos ( 7 x ) + 165 cos ( 5 x ) + 330 cos ( 3 x ) + 462 cos ( x ) ] cos 12 x = 1 2048 [ cos ( 12 x ) + 12 cos ( 10 x ) + 66 cos ( 8 x ) + 220 cos ( 6 x ) + 495 cos ( 4 x ) + 792 cos ( 2 x ) + 462 ] {\displaystyle {\begin{cases}\cos ^{1}x&=&\left[\cos x\right]\\\cos ^{2}x&={\frac {1}{2}}&\left[\cos \left(2x\right)+1\right]\\\cos ^{3}x&={\frac {1}{4}}&\left[\cos \left(3x\right)+3\cos \left(x\right)\right]\\\cos ^{4}x&={\frac {1}{8}}&\left[\cos \left(4x\right)+4\cos \left(2x\right)+3\right]\\\cos ^{5}x&={\frac {1}{16}}&\left[\cos \left(5x\right)+5\cos \left(3x\right)+10\cos \left(x\right)\right]\\\cos ^{6}x&={\frac {1}{32}}&\left[\cos \left(6x\right)+6\cos \left(4x\right)+15\cos \left(2x\right)+10\right]\\\cos ^{7}x&={\frac {1}{64}}&\left[\cos \left(7x\right)+7\cos \left(5x\right)+21\cos \left(3x\right)+35\cos \left(x\right)\right]\\\cos ^{8}x&={\frac {1}{128}}&\left[\cos \left(8x\right)+8\cos \left(6x\right)+28\cos \left(4x\right)+56\cos \left(2x\right)+35\right]\\\cos ^{9}x&={\frac {1}{256}}&\left[\cos \left(9x\right)+9\cos \left(7x\right)+36\cos \left(5x\right)+84\cos \left(3x\right)+126\cos \left(x\right)\right]\\\cos ^{10}x&={\frac {1}{512}}&\left[\cos \left(10x\right)+10\cos \left(8x\right)+45\cos \left(6x\right)+120\cos \left(4x\right)+210\cos \left(2x\right)+126\right]\\\cos ^{11}x&={\frac {1}{1024}}&\left[\cos \left(11x\right)+11\cos \left(9x\right)+55\cos \left(7x\right)+165\cos \left(5x\right)+330\cos \left(3x\right)+462\cos \left(x\right)\right]\\\cos ^{12}x&={\frac {1}{2048}}&\left[\cos \left(12x\right)+12\cos \left(10x\right)+66\cos \left(8x\right)+220\cos \left(6x\right)+495\cos \left(4x\right)+792\cos \left(2x\right)+462\right]\end{cases}}}
{ sin 1 x = [ sin x ] sin 2 x = 1 2 [ − cos ( 2 x ) + 1 ] sin 3 x = 1 4 [ − sin ( 3 x ) + 3 sin ( x ) ] sin 4 x = 1 8 [ cos ( 4 x ) − 4 cos ( 2 x ) + 3 ] sin 5 x = 1 16 [ sin ( 5 x ) − 5 sin ( 3 x ) + 10 sin ( x ) ] sin 6 x = 1 32 [ − cos ( 6 x ) + 6 cos ( 4 x ) − 15 cos ( 2 x ) + 10 ] sin 7 x = 1 64 [ − sin ( 7 x ) + 7 sin ( 5 x ) − 21 sin ( 3 x ) + 35 sin ( x ) ] sin 8 x = 1 128 [ cos ( 8 x ) − 8 cos ( 6 x ) + 28 cos ( 4 x ) − 56 cos ( 2 x ) + 35 ] sin 9 x = 1 256 [ sin ( 9 x ) − 9 sin ( 7 x ) + 36 sin ( 5 x ) − 84 sin ( 3 x ) + 126 sin ( x ) ] sin 10 x = 1 512 [ − cos ( 10 x ) + 10 cos ( 8 x ) − 45 cos ( 6 x ) + 120 cos ( 4 x ) − 210 cos ( 2 x ) + 126 ] sin 11 x = 1 1024 [ − sin ( 11 x ) + 11 sin ( 9 x ) − 55 sin ( 7 x ) + 165 sin ( 5 x ) − 330 sin ( 3 x ) + 462 sin ( x ) ] sin 12 x = 1 2048 [ cos ( 12 x ) − 12 cos ( 10 x ) + 66 cos ( 8 x ) − 220 cos ( 6 x ) + 495 cos ( 4 x ) − 792 cos ( 2 x ) + 462 ] {\displaystyle {\begin{cases}\sin ^{1}x&=&\left[\sin x\right]\\\sin ^{2}x&={\frac {1}{2}}&\left[-\cos \left(2x\right)+1\right]\\\sin ^{3}x&={\frac {1}{4}}&\left[-\sin \left(3x\right)+3\sin \left(x\right)\right]\\\sin ^{4}x&={\frac {1}{8}}&\left[\cos \left(4x\right)-4\cos \left(2x\right)+3\right]\\\sin ^{5}x&={\frac {1}{16}}&\left[\sin \left(5x\right)-5\sin \left(3x\right)+10\sin \left(x\right)\right]\\\sin ^{6}x&={\frac {1}{32}}&\left[-\cos \left(6x\right)+6\cos \left(4x\right)-15\cos \left(2x\right)+10\right]\\\sin ^{7}x&={\frac {1}{64}}&\left[-\sin \left(7x\right)+7\sin \left(5x\right)-21\sin \left(3x\right)+35\sin \left(x\right)\right]\\\sin ^{8}x&={\frac {1}{128}}&\left[\cos \left(8x\right)-8\cos \left(6x\right)+28\cos \left(4x\right)-56\cos \left(2x\right)+35\right]\\\sin ^{9}x&={\frac {1}{256}}&\left[\sin \left(9x\right)-9\sin \left(7x\right)+36\sin \left(5x\right)-84\sin \left(3x\right)+126\sin \left(x\right)\right]\\\sin ^{10}x&={\frac {1}{512}}&\left[-\cos \left(10x\right)+10\cos \left(8x\right)-45\cos \left(6x\right)+120\cos \left(4x\right)-210\cos \left(2x\right)+126\right]\\\sin ^{11}x&={\frac {1}{1024}}&\left[-\sin \left(11x\right)+11\sin \left(9x\right)-55\sin \left(7x\right)+165\sin \left(5x\right)-330\sin \left(3x\right)+462\sin \left(x\right)\right]\\\sin ^{12}x&={\frac {1}{2048}}&\left[\cos \left(12x\right)-12\cos \left(10x\right)+66\cos \left(8x\right)-220\cos \left(6x\right)+495\cos \left(4x\right)-792\cos \left(2x\right)+462\right]\end{cases}}}
Ainsi :
{ cos ( 1 x ) = cos x cos ( 2 x ) = 2 cos 2 x − 1 cos ( 3 x ) = 4 cos 3 x − 3 cos x cos ( 4 x ) = 8 cos 4 x − 8 cos 2 x + 1 cos ( 5 x ) = 16 cos 5 x − 20 cos 3 x + 5 cos x cos ( 6 x ) = 32 cos 6 x − 48 cos 4 x + 18 cos 2 x − 1 cos ( 7 x ) = 64 cos 7 x − 112 cos 5 x + 56 cos 3 x − 7 cos x cos ( 8 x ) = 128 cos 8 x − 256 cos 6 x + 160 cos 4 x − 32 cos 2 x + 1 cos ( 9 x ) = 256 cos 9 x − 576 cos 7 x + 432 cos 5 x − 120 cos 3 x + 9 cos x cos ( 10 x ) = 512 cos 10 x − 1280 cos 8 x + 1120 cos 6 x − 400 cos 4 x + 50 cos 2 x − 1 cos ( 11 x ) = 1024 cos 11 x − 2816 cos 9 x + 2816 cos 7 x − 1232 cos 5 x + 220 cos 3 x − 11 cos x cos ( 12 x ) = 2048 cos 12 x − 6144 cos 10 x + 6912 cos 8 x − 3584 cos 6 x + 840 cos 4 x − 72 cos 2 x + 1 {\displaystyle {\begin{cases}\cos \left(1x\right)&=\cos x\\\cos \left(2x\right)&=2\cos ^{2}x-1\\\cos \left(3x\right)&=4\cos ^{3}x-3\cos x\\\cos \left(4x\right)&=8\cos ^{4}x-8\cos ^{2}x+1\\\cos \left(5x\right)&=16\cos ^{5}x-20\cos ^{3}x+5\cos x\\\cos \left(6x\right)&=32\cos ^{6}x-48\cos ^{4}x+18\cos ^{2}x-1\\\cos \left(7x\right)&=64\cos ^{7}x-112\cos ^{5}x+56\cos ^{3}x-7\cos x\\\cos \left(8x\right)&=128\cos ^{8}x-256\cos ^{6}x+160\cos ^{4}x-32\cos ^{2}x+1\\\cos \left(9x\right)&=256\cos ^{9}x-576\cos ^{7}x+432\cos ^{5}x-120\cos ^{3}x+9\cos x\\\cos \left(10x\right)&=512\cos ^{10}x-1280\cos ^{8}x+1120\cos ^{6}x-400\cos ^{4}x+50\cos ^{2}x-1\\\cos \left(11x\right)&=1024\cos ^{11}x-2816\cos ^{9}x+2816\cos ^{7}x-1232\cos ^{5}x+220\cos ^{3}x-11\cos x\\\cos \left(12x\right)&=2048\cos ^{12}x-6144\cos ^{10}x+6912\cos ^{8}x-3584\cos ^{6}x+840\cos ^{4}x-72\cos ^{2}x+1\end{cases}}}
{ sin ( 1 x ) = sin x sin ( 2 x ) = ( 2 sin x ) cos x sin ( 3 x ) = − 4 sin 3 x + 3 sin x sin ( 4 x ) = ( − 8 sin 3 x + 4 sin x ) cos x sin ( 5 x ) = 16 sin 5 x − 20 sin 3 x + 5 sin x sin ( 6 x ) = ( 32 sin 5 x − 32 sin 3 x + 6 sin x ) cos x sin ( 7 x ) = − 64 sin 7 x + 112 sin 5 x − 56 sin 3 x + 7 sin x sin ( 8 x ) = ( − 128 sin 7 x + 192 sin 5 x − 80 sin 3 x + 8 sin x ) cos x sin ( 9 x ) = 256 sin 9 x − 576 sin 7 x + 432 sin 5 x − 120 sin 3 x + 9 sin x sin ( 10 x ) = ( 512 sin 9 x − 1024 sin 7 x + 672 sin 5 x − 160 sin 3 x + 10 sin x ) cos x sin ( 11 x ) = − 1024 sin 11 x + 2816 sin 9 x − 2816 sin 7 x + 1232 sin 5 x − 220 sin 3 x + 11 sin x sin ( 12 x ) = ( − 2048 sin 11 x + 5120 sin 9 x − 4608 sin 7 x + 1792 sin 5 x − 280 sin 3 x + 12 sin x ) cos x {\displaystyle {\begin{cases}\sin \left(1x\right)&=\sin x\\\sin \left(2x\right)&=\left(2\sin x\right)\cos x\\\sin \left(3x\right)&=-4\sin ^{3}x+3\sin x\\\sin \left(4x\right)&=\left(-8\sin ^{3}x+4\sin x\right)\cos x\\\sin \left(5x\right)&=16\sin ^{5}x-20\sin ^{3}x+5\sin x\\\sin \left(6x\right)&=\left(32\sin ^{5}x-32\sin ^{3}x+6\sin x\right)\cos x\\\sin \left(7x\right)&=-64\sin ^{7}x+112\sin ^{5}x-56\sin ^{3}x+7\sin x\\\sin \left(8x\right)&=\left(-128\sin ^{7}x+192\sin ^{5}x-80\sin ^{3}x+8\sin x\right)\cos x\\\sin \left(9x\right)&=256\sin ^{9}x-576\sin ^{7}x+432\sin ^{5}x-120\sin ^{3}x+9\sin x\\\sin \left(10x\right)&=\left(512\sin ^{9}x-1024\sin ^{7}x+672\sin ^{5}x-160\sin ^{3}x+10\sin x\right)\cos x\\\sin \left(11x\right)&=-1024\sin ^{11}x+2816\sin ^{9}x-2816\sin ^{7}x+1232\sin ^{5}x-220\sin ^{3}x+11\sin x\\\sin \left(12x\right)&=\left(-2048\sin ^{11}x+5120\sin ^{9}x-4608\sin ^{7}x+1792\sin ^{5}x-280\sin ^{3}x+12\sin x\right)\cos x\end{cases}}}
![{\displaystyle {\begin{cases}\cos ^{1}x&=&\left[\cos x\right]\\\cos ^{2}x&={\frac {1}{2}}&\left[\cos \left(2x\right)+1\right]\\\cos ^{3}x&={\frac {1}{4}}&\left[\cos \left(3x\right)+3\cos \left(x\right)\right]\\\cos ^{4}x&={\frac {1}{8}}&\left[\cos \left(4x\right)+4\cos \left(2x\right)+3\right]\\\cos ^{5}x&={\frac {1}{16}}&\left[\cos \left(5x\right)+5\cos \left(3x\right)+10\cos \left(x\right)\right]\\\cos ^{6}x&={\frac {1}{32}}&\left[\cos \left(6x\right)+6\cos \left(4x\right)+15\cos \left(2x\right)+10\right]\\\cos ^{7}x&={\frac {1}{64}}&\left[\cos \left(7x\right)+7\cos \left(5x\right)+21\cos \left(3x\right)+35\cos \left(x\right)\right]\\\cos ^{8}x&={\frac {1}{128}}&\left[\cos \left(8x\right)+8\cos \left(6x\right)+28\cos \left(4x\right)+56\cos \left(2x\right)+35\right]\\\cos ^{9}x&={\frac {1}{256}}&\left[\cos \left(9x\right)+9\cos \left(7x\right)+36\cos \left(5x\right)+84\cos \left(3x\right)+126\cos \left(x\right)\right]\\\cos ^{10}x&={\frac {1}{512}}&\left[\cos \left(10x\right)+10\cos \left(8x\right)+45\cos \left(6x\right)+120\cos \left(4x\right)+210\cos \left(2x\right)+126\right]\\\cos ^{11}x&={\frac {1}{1024}}&\left[\cos \left(11x\right)+11\cos \left(9x\right)+55\cos \left(7x\right)+165\cos \left(5x\right)+330\cos \left(3x\right)+462\cos \left(x\right)\right]\\\cos ^{12}x&={\frac {1}{2048}}&\left[\cos \left(12x\right)+12\cos \left(10x\right)+66\cos \left(8x\right)+220\cos \left(6x\right)+495\cos \left(4x\right)+792\cos \left(2x\right)+462\right]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4809ce35c6c9a5f188619ca1330435d7668336a5)
![{\displaystyle {\begin{cases}\sin ^{1}x&=&\left[\sin x\right]\\\sin ^{2}x&={\frac {1}{2}}&\left[-\cos \left(2x\right)+1\right]\\\sin ^{3}x&={\frac {1}{4}}&\left[-\sin \left(3x\right)+3\sin \left(x\right)\right]\\\sin ^{4}x&={\frac {1}{8}}&\left[\cos \left(4x\right)-4\cos \left(2x\right)+3\right]\\\sin ^{5}x&={\frac {1}{16}}&\left[\sin \left(5x\right)-5\sin \left(3x\right)+10\sin \left(x\right)\right]\\\sin ^{6}x&={\frac {1}{32}}&\left[-\cos \left(6x\right)+6\cos \left(4x\right)-15\cos \left(2x\right)+10\right]\\\sin ^{7}x&={\frac {1}{64}}&\left[-\sin \left(7x\right)+7\sin \left(5x\right)-21\sin \left(3x\right)+35\sin \left(x\right)\right]\\\sin ^{8}x&={\frac {1}{128}}&\left[\cos \left(8x\right)-8\cos \left(6x\right)+28\cos \left(4x\right)-56\cos \left(2x\right)+35\right]\\\sin ^{9}x&={\frac {1}{256}}&\left[\sin \left(9x\right)-9\sin \left(7x\right)+36\sin \left(5x\right)-84\sin \left(3x\right)+126\sin \left(x\right)\right]\\\sin ^{10}x&={\frac {1}{512}}&\left[-\cos \left(10x\right)+10\cos \left(8x\right)-45\cos \left(6x\right)+120\cos \left(4x\right)-210\cos \left(2x\right)+126\right]\\\sin ^{11}x&={\frac {1}{1024}}&\left[-\sin \left(11x\right)+11\sin \left(9x\right)-55\sin \left(7x\right)+165\sin \left(5x\right)-330\sin \left(3x\right)+462\sin \left(x\right)\right]\\\sin ^{12}x&={\frac {1}{2048}}&\left[\cos \left(12x\right)-12\cos \left(10x\right)+66\cos \left(8x\right)-220\cos \left(6x\right)+495\cos \left(4x\right)-792\cos \left(2x\right)+462\right]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5bfd9d273e2d8c95bdc8955de32708470a294d)
