Cos^2 (x) + cos^2 (Pi/6) = cos^2 (2x) + sin^2 (Pi/3)

Cos^2 (x) + cos^2 (Pi/6) = cos^2 (2 * x) + sin^2 (pi/3);

cos^2 x + √3/2 * √3/2 = cos^2 (2 * x) + √3/2 * √3/2;

cos^2 x + √9/4 = cos^2 (2 * x) + √9/4;

cos^2 x + 3/4 = cos^2 (2 * x) + 3/4;

Приведем подобные.

cos^2 x = cos^2 (2 * x);

cos^2 x - cos^2 (2 * x) = 0;

cos^2 x - (cos^2 x - sin^2 x) ^2 = 0;

cos^2 x - (cos^2 x - (1 - cos^2 x)) ^2 = 0;

cos^2 x - (2 * cos^2 - 1) ^2 = 0;

Разложим на множители:

(cos x + 2 * cos^2 x - 1) * (cos x - 2 * cos^2 x + 1) = 0;

- (2 * cos^2 x + cos x - 1) * (2 * cos^2 x - cos x - 1) = 0;

1) 2 * cos^2 x + cos x - 1 = 0;

D = 1 - 4 * 2 * (-1) = 9;

cos x = (-1 + 3) / 6 = 1/3;

x = + - arccos (1/3) + 2 * pi * n;

cos x = - 4/6 = - 2/3;

x = + - arccos (-2/3) + 2 * pi * n;

2) 2 * cos^2 x - cos x - 1 = 0;

D = 1 - 4 * 2 * (-1) = 9;

cos x = (1 + 3) / 4 = 1;

x = 2 * pi * n;

cos x = - 2/4 = - 1/2;

x = + -2 * pi/3 + 2 * pi * n.