verify the identity

Question

sin x(1 - 2 cos^2 x + cos^4 x) = sin^5 x

1-2cos^2 x looks like 2cos^2x-1 its just backwards. I am not sure where to start.

Answer 1

good observation
recall that cos 2A = 2cos^2 A - 1 = 1 - 2sin^2 A

so 1 - 2 cos^2 x = 2sin^2 x - 1

LS
=sin x(1 - 2 cos^2 x + cos^4 x)
= sinx(2sin^2 x - 1 + (cos^2x)(cos^2x)
= sinx( 2 sin^2x - 1 + (1-sin^2x)(1 -sin^2x))
= sinx(2sin^2x - 1 + 1 - 2sin^2x + sin^4x)
= sinx(sin^4x)
= sin^5 x
= RS