Если sinx+cosx=a то чему равен tg^3x+ctg^3x?

1) Sin x + Cos x = a |²

Sin²x + 2Sin xCos x + Cos²x = a²

1 + 2Sin xCos x = a²

2Sin xCos x = a² - 1⇒ Sin x Cos x = (a² - 1) / 2

2) tg³ x + Ctg³x = (tg x + Ctg x) (tg²x - tg xCtg x + Ctg²x) =

=1 / (Sin xCos x) · (Sin²x/Cos²x - 1 + Cos²x/Sin² x) =

=1 / (Sin xCos x) · ((Sin^4x + Cos^4x) / Cos²xSin²x - 1) =

= 1 / (Sin xCos x) · ((Sin^4x + Cos^4 x + 2Sin²xCos²x - 2Sin²xCos²x / (Sin² xCos² x) - 1) = 1 / (Sin xCos x) · ((Sin²x + Cos²x) ² - 2Sin²xCos²x / (Sin² xCos² x) - 1) = 1 / (Sin xCos x) · ((1 - 2Sin²xCos²x) / (Sin² xCos² x) - 1) =

Теперь можно заменить Sin xCos x и упростить