Verified answer

The first thing you should notice is that sin x cos^2 x is a factor of both sides, so you can factor that out and just deal with what's left.

The only other identity you need to know in this case is Pythagoras' theorem, which states that:

cos² x + sin² x = 1

This can be rearranged into a bunch of identities. In your case:

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

So:

sin^5 x cos^2 x

= sin x cos^2 x sin^4 x

= sin x cos^2 x (1 - cos^2 x)^2

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

= sin x (cos^2 x - 2 cos^4 x + cos^6 x)