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.