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.