Докажите тождество: (cosx + cos x/2)^2 + (sinx + sin x/2)^2 = 2sin x/2· ctg x/4

Question

Докажите тождество:
(cosx + cos x/2)^2 + (sinx + sin x/2)^2 = 2sin x/2· ctg x/4

Answer 1

Cos²x+2cosxcos(x/2)+cos²(x/2)+sin²x+2sinxsin(x/2)+sin²(x/2)=
=2+2cos(x-x/2)=2+2cosx/2=2(1+cosx/2)
2sinx/2*ctgx/4=2sinx/4*cosx/4*(cosx/4)/(sinx/4)=2cos²x/4=1+cosx/2