sin^3 x-cos^3 x=1+ (sin2x/2)

Левая часть. Разность кубов
sin^3 x - cos^3 x = (sin x - cos x)(sin^2 x + sin x*cos x + cos^2 x)
Правая часть
1 + sin 2x / 2 = sin^2 x + cos^2 x + sin x*cos x
Получаем
(sin x-cos x)(sin^2 x+sin x*cos x+cos^2 x) = sin^2 x+cos^2 x+sin x*cos x
(sin^2 x + cos^2 x + sin x*cos x)(sin x - cos x - 1) = 0
1) sin^2 x + cos^2 x + sin x*cos x = 0 
1 + sin 2x / 2 = 0
sin 2x = -2 - решений нет
2) sin x - cos x - 1 = 0
2sin(x/2)*cos(x/2) - cos^2(x/2) + sin^2(x/2) - cos^2(x/2) - sin^2(x/2) = 0
2sin(x/2)*cos(x/2) - 2cos^2(x/2) = 0
2cos(x/2)*(sin(x/2) - cos(x/2)) = 0
cos(x/2) = 0; x/2 = pi/2 + pi*k;
x1 = pi + 2pi*k

sin(x/2) - cos(x/2) = 0
sin(x/2) = cos(x/2)
tg(x/2) = 1; x/2 = pi/4 + pi*k;
x2 = pi/2 + 2pi*k