Sin^6 (x) = 1/4-cos^6 (x)

1. Сумма кубов двух выражений:

sin^6 (x) = 1/4 - cos^6 (x); sin^6 (x) + cos^6 (x) = 1/4; (sin^2 (x)) ^3 + (cos^2 (x)) ^3 = 1/4; (sin^2 (x) + cos^2 (x)) (sin^4 (x) - sin^2 (x) * cos^2 (x) + cos^4 (x)) = 1/4; sin^4 (x) - sin^2 (x) * cos^2 (x) + cos^4 (x) = 1/4.

2. Выделим квадрат двучлена:

(sin^2 (x) + cos^2 (x)) ^2 - 3sin^2 (x) * cos^2 (x) = 1/4; 1 - 3sin^2 (x) * cos^2 (x) = 1/4; 3sin^2 (x) * cos^2 (x) = 3/4; sin^2 (x) * cos^2 (x) = 1/4; sinx * cosx = ±1/2; 2sinx * cosx = ±1; sin (2x) = ±1; 2x = π/2 + πk, k ∈ Z; x = π/4 + πk/2, k ∈ Z.

Ответ: π/4 + πk/2, k ∈ Z.