Cos^6+sin^6=15/8cos2x-1/2

1. Сумма кубов:

cos^6 (x) + sin^6 (x) = 15/8 * cos2x - 1/2; (cos^2 (x) + sin^2 (x)) (cos^4 (x) - cos^2 (x) * sin^2 (x) + sin^4 (x)) = 15/8 * cos2x - 1/2; cos^4 (x) - cos^2 (x) * sin^2 (x) + sin^4 (x) = 15/8 * cos2x - 1/2.

2. Выделим полный квадрат:

(sin^2 (x) + cos^2 (x)) ^2 - 3sin^2 (x) cos^2 (x) = 15/8 * cos2x - 1/2; 1 - 3sin^2 (x) cos^2 (x) = 15/8 * cos2x - 1/2; 3/2 - 3sin^2 (x) cos^2 (x) = 15/8 * cos2x.

3. Умножим на 8/3:

4 - 8sin^2 (x) cos^2 (x) = 5cos2x; 4 - 2 (2sinx * cosx) ^2 = 5cos2x; 2 + 2 - 2sin^2 (2x) = 5cos2x; 2 + 2cos^2 (2x) = 5cos2x; 2cos^2 (2x) - 5cos2x + 2 = 0; D = 5^2 - 4 * 2 * 2 = 25 - 16 = 9; cos2x = (5 ± √9) / 4 = (5 ± 3) / 4.

a) cos2x = (5 - 3) / 4 = 2/4 = 1/2;

2x = ±π/3 + 2πk, k ∈ Z; x = ±π/6 + πk, k ∈ Z.

b) cos2x = (5 + 3) / 4 = 8/4 = 2, нет решений.

Ответ: ±π/6 + πk, k ∈ Z.