Sin^6 x + cos^6 x = cos 2x

   Будут применены последующие формулы:

• (a + b)2 = a^2 + 2ab + b^2;
• a^3 + b^3 = (a + b)(a^2 - ab + b^2);
• sin^2 + cos^2 = 1;
• 1 - 2sin^2 = cos2;

• sin^6x + cos^6x = cos2x;
• (sin^2x)^3 + (cos^2x)^3 = cos2x;
• (sin^2x + cos^2x)(sin^4x - sin^2x * cos^2x + cos^4x) = cos2x;
• (sin^2x + cos^2x)^2 - 3sin^2x = cos2x;
• 1 - 3sin^2x = cos2x;
• 2 - 6sin^2x = 2cos2x;
• 3 - 6sin^2x - 1 = 2cos2x;
• 3(1 - 2sin^2x) - 1 = 2cos2x;
• 3cos2x - 1 = 2cos2x;
• 3cos2x - 2cos2x = 1;
• cos2x = 1;
• 2x = 2k, k Z;
• x = k, k Z.

   Ответ: k, k Z.