Sin^3 (x) + cos^3 (x) =, если sin (x) + cos (x) = a

X^3 + y^3 = (x+y) * (x^2 - x*y + y^2)

sin^3 (x) + cos^3 (x) = (sinx + cosx) * (sin^2 (x) - sinx*cosx + cos^2 (x))

sin^2 (x) + cos^2 (x) = 1 - основное тригонометрич. тождество

(sinx + cosx) * (1 - sinx*cosx)

(sinx + cosx) ^2 = sin^2 (x) + 2sinx*cosx + cos^2x = 1 + 2sinx*cosx = 1 + 2a

2sinx*cosx = 2a + 1 - 1, sinx*cosx = a

(sinx + cosx) * (1 - sinx*cosx) = a * (1 - a) = a - a^2