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