cos4x+cos2x=sin9x+sin3x

1. Воспользуемся тригонометрическими формулами:

sina + sinb = 2sin ((a + b) / 2) * cos ((a - b) / 2); sina - sinb = 2sin ((a - b) / 2) * cos ((a + b) / 2); cosa + cosb = 2cos ((a + b) / 2) * cos ((a - b) / 2); cos4x + cos2x = sin9x + sin3x; 2cos3x * cosx = 2sin6x * cos3x; 2cos3x * cosx - 2sin6x * cos3x = 0; 2cos3x (cosx - sin6x) = 0; 2cos3x (sin (x + π/2) - sin6x) = 0; cos3x (sin6x - sin (x + π/2)) = 0; cos3x * 2sin (5x/2 - π/4) * cos (7x/2 + π/4) = 0.

2. Приравняем к нулю каждый из множителей:

[cos3x = 0;

[sin (5x/2 - π/4) = 0;

[cos (7x/2 + π/4) = 0; [3x = π/2 + πk;

[5x/2 - π/4 = πk;

[7x/2 + π/4 = π/2 + πk; [x = π/6 + πk/3;

[5x = π/2 + 2πk;

[7x = π/2 + 2πk; [x = π/6 + πk/3;

[x = π/10 + 2πk/5;

[x = π/14 + 2πk/7.

Ответ: π/6 + πk/3; π/10 + 2πk/5; π/14 + 2πk/7, k ∈ Z.