Question

Rewrite sin^4(x/2) in terms of the first power of cosine

Answer 1

sin^4 (x/2) = (sin^2 x/2)(sin^2 x/2) = (1 - cos^2 x/2)(1 - cos^2 x/2)
= cos^4 x/2 - 2 cos^2 x/2 + 1 =
Replace cos^4 x/2 = cos ^2 x/2*cos^2 x/2 = (1/4)(1 + cos x)^2
Replace -2cos^2 x/2 = -(1 + cos x)
Finally: sin^4 x/2 = (1/4) (1 + cos x)^2 - cos x