I have to verify this identity algebraically:

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

So I have to make the left side equal the right,

My work so far:

sin x (sin^2 x cos^2 x + 4 coos x)

sin x (sin^2 x [1 - sin^2 x] + 4 cos^4 x)

sin x ([sin^2 x - sin^4 x] + 4 cos^4 x)

sin x ([sin^2 x + cos^4 x + 1] + 4 cos^4 x)

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

Now I am stuck. Am I correct so far? If I am, what do I do next?

Any help is appreciated.