Sin^4x+cos^4x=cos^2 2 x

Решим уравнение Sin^4 x + cos^4 x = cos^2 x. 

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

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

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

2 * (cos^2 x)^2 - 3 * cos^2 x + 1 = 0; 

Пусть cos^2 x = a, тогда: 

2 * a^2 - 3 * a + 1 = 0; 

a = 0.5; 

a = 1; 

Тогда: 

1) cos^2 x = 1/2; 

(cos x - √2/2) * (cos x + √2/2) = 0; 

1. cos x = √2/2; 

x = +- pi/4 + 2 * pi * n; 

2. cos x = -√2/2; 

x = +-3 * pi/4 + 2 * pi * n; 

2) cos^2 x = 1; 

cos^2 x - 1 = 0; 

(cos x - 1) * (cos x + 1) = 0; 

1. cos x = 1; 

x = 2 * pi * n; 

2. cos x = -1; 

x = pi + 2 * pi * n, где n принадлежит Z.