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.