use power reducing identities to prove the identity

Question

sin^4x=1/8(3-4cos2x+cos4x)

cos^3x=(1/2cosx) (1+cos2x)

thanks :)

Answer 1

cos 2x = 2cos^2 x - 1
so, 1/2 cos x (1+2cos^2 x - 1) = cos^3 x

cos 4x = 1 - 2sin^2 2x
= 1 - 8sin^2 x cos^2 x
= 1 - 8sin^2 x (1 - sin^2 x)
= 1 - 8sin^2 x + 8 sin^4 x

cos 2x = 1 - 2sin^2 x
4cos 2x = 4 - 8sin^2 x

1/8(3-4cos2x+cos4x)
= 1/8(3 - 4 + 8sin^2 x + 1 - 8sin^2 x + 8 sin^4 x)
= 1/8(8sin^4 x)
= sin^4 x