Sina+cosa=a sin^3a+cos^3a=?

Question

Sina+cosa=a
sin^3a+cos^3a=?

Answer 1

Sina+cosa=a
(sina+cosa)²=sin²a+2sinacosa+cos²a=1+2sinacosa=a²⇒sinacosa=(a²-1)/2
sin³a+cos³a=(sina+cosa)(sin²a-sinacosa+cos²a)=(sina+cosa)(1-sinacosa)
=a*(1-(a²-1)/2)=a*(2-a²+1)/2=a(3-a²)/2