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