Cos^2 (3x) + cos^2 (4x) + cos^2 (5x) = 3/2

1. Умножим обе части уравнения на 2:

cos^2 (3x) + cos^2 (4x) + cos^2 (5x) = 3/2; 2cos^2 (3x) + 2cos^2 (4x) + 2cos^2 (5x) = 3; 2cos^2 (3x) - 1 + 2cos^2 (4x) - 1 + 2cos^2 (5x) - 1 = 0.

2. Косинус двойного угла и сумма косинусов:

cos6x + cos8x + cos10x = 0; 2cos ((10x + 6x) / 2) * cos ((10x - 6x) / 2) + cos8x = 0; 2cos8x * cos2x + cos8x = 0; cos8x (2cos2x + 1) = 0; [cos8x = 0;

[2cos2x + 1 = 0; [cos8x = 0;

[cos2x = - 1/2; [8x = π/2 + πk, k ∈ Z;

[2x = ±2π/3 + 2πk, k ∈ Z; [x = π/16 + πk/8, k ∈ Z;

[x = ±π/3 + πk, k ∈ Z.

Ответ: π/16 + πk/8; ±π/3 + πk, k ∈ Z.