cos 5x = cos (4x+x) = cos 4x cos x – sin 4x sin x = (2 cos ^2 (2x) - 1) cos x - 2 sin 2x cos 2x sin x = (2 (2 cos ^2 x - 1) ^2 – 1) cos x – 2 (2 sin x cos x) (2 cos ^2 x – 1) sin x = (2 (2 cos ^2 x – 1) ^2-1) cos x -4 sin ^2 x cos x (2 cos ^2 x – 1) = (2 (4 cos ^4 x – 4 cos ^2 x + 1) -1) cos x – 4 (1 – cos ^2 x) cos x (2 cos ^2 x – 1) = (8 cos ^4 x – 8 cos ^2 x +2 -1) cos x – 4 (cos x – cos ^3 x) (2 cos ^2 x – 1) = 8 cos ^5 x – 8 cos ^3 x + cos x – 8 cos ^3 x + 8 cos ^5 x + 4 cos x – 4 cos ^3 x = 16 cos ^5 x – 20 cos ^3 x+ 5 cos x
Используются тождества:
cos (x + y) = cos x cos y – sin x sin y
sin 2x = 2 sin x cos x
cos 2x = 2 cos ^2 x – 1
cos 5x = cos (4x+x) = cos 4x cos x – sin 4x sin x = (2 cos ^2 (2x) - 1) cos x - 2 sin 2x cos 2x sin x = (2 (2 cos ^2 x - 1) ^2 – 1) cos x – 2 (2 sin x cos x) (2 cos ^2 x – 1) sin x = (2 (2 cos ^2 x – 1) ^2-1) cos x -4 sin ^2 x cos x (2 cos ^2 x – 1) = (2 (4 cos ^4 x – 4 cos ^2 x + 1) -1) cos x – 4 (1 – cos ^2 x) cos x (2 cos ^2 x – 1) = (8 cos ^4 x – 8 cos ^2 x +2 -1) cos x – 4 (cos x – cos ^3 x) (2 cos ^2 x – 1) = 8 cos ^5 x – 8 cos ^3 x + cos x – 8 cos ^3 x + 8 cos ^5 x + 4 cos x – 4 cos ^3 x = 16 cos ^5 x – 20 cos ^3 x+ 5 cos x
Есть еще вариант решения - разложением