Решите уравнение cos^6(pi/4-x) + cos^6(pi/4+x) =o,5 60

Во-первых, разложим косинусы в сумму
cos(pi/4 - x) = cos(pi/4)*cos(x) + sin(pi/4)*sin(x) =
= 1/√2*cos x + 1/√2*sin x = 1/√2*(cos x + sin x)
cos(pi/4 + x) = cos(pi/4)*cos(x) - sin(pi/4)*sin(x) =
= 1/√2*cos x - 1/√2*sin x = 1/√2*(cos x - sin x)
Подставляем
cos^6(pi/4-x) + cos^6(pi/4+x) = 1/2^3*(cos x+sin x)^6 + 1/2^3*(cos x-sin x)^6 =
= 1/8*[(cos x+sin x)^6 + (cos x-sin x)^6] = 0,5
(cos x + sin x)^6 + (cos x - sin x)^6 = 4
Во-вторых, разложим сумму кубов
[(cos x + sin x)^2 + (cos x - sin x)^2] * [(cos x + sin x)^4 - 
- (cos x + sin x)^2*(cos x - sin x)^2 + (cos x - sin x)^4] = 4
Первая скобка
cos^2 x + 2cos x*sin x + sin^2 x + cos^2 x - 2cos x*sin x + sin^2 x =
= (cos^2 x + sin^2 x) + (2cos x*sin x - 2cos x*sin x) + (cos^2 x + sin^2 x) = 2
Вторая скобка
(cos x+sin x)^4 - (cos x+sin x)^2*(cos x-sin x)^2 + (cos x-sin x)^4 =
= ((cos x+sin x)^2)^2 + ((cos x-sin x)^2)^2 - (cos x+sin x)^2*(cos x-sin x)^2 =
= (1+2cos x*sin x)^2 + (1-2cos x*sin x)^2 - (1+2cos x*sin x)(1-2cos x*sin x) =
= (1+sin 2x)^2 + (1-sin 2x)^2 - (1-sin^2 2x) = 1 + 2sin 2x + sin^2 2x +
+ 1 - 2sin 2x + sin^2 2x - 1 + sin^2 2x = 1 + 3sin^2 2x
Подставляем в уравнение
2(1 + 3sin^2 2x) = 4
1 + 3sin^2 2x = 2
3sin^2 2x = 1
sin^2 2x = 1/3
sin 2x = 1/√3
2x = (-1)^n*arcsin(1/√3) + pi*k
x = (-1)^n*1/2*arcsin(1/√3) + pi/2*k