Решить уравнение cos12x=cos6x+sin6x

Воспользуемся формулой двойного угла косинус:

cos (2a) = cos^2 (a) - sin^2 (a);

cos (12x) = cos (6x) + sin (6x); cos^2 (6x) - sin^2 (6x) - (cos (6x) + sin (6x)) = 0; (cos (6x) + sin (6x)) (cos (6x) - sin (6x)) - (cos (6x) + sin (6x)) = 0; (cos (6x) + sin (6x)) (cos (6x) - sin (6x) - 1) = 0;

[cos (6x) + sin (6x) = 0;

[cos (6x) - sin (6x) - 1 = 0.

a) cos (6x) + sin (6x) = 0;

sin (6x) = - cos (6x); tg (6x) = - 1; 6x = - π/4 + πk, k ∈ Z; x = - π/24 + πk/6, k ∈ Z.

b) cos (6x) - sin (6x) - 1 = 0;

cos (6x) - sin (6x) = 1; √2/2 * cos (6x) - √2/2 * sin (6x) = √2/2; cos (π/4) * cos (6x) - sin (π/4) * sin (6x) = √2/2; cos (6x + π/4) = √2/2; 6x + π/4 = ±π/4 + 2πk, k ∈ Z; 6x = - π/4 ± π/4 + 2πk, k ∈ Z; [6x = - π/4 - π/4 + 2πk, k ∈ Z;

[6x = - π/4 + π/4 + 2πk, k ∈ Z; [6x = - π/2 + 2πk, k ∈ Z;

[6x = 2πk, k ∈ Z; [x = - π/12 + πk/3, k ∈ Z;

[x = πk/3, k ∈ Z.

Ответ: - π/24 + πk/6; - π/12 + πk/3; πk/3, k ∈ Z.