It is quite simple just write sin^2 (x) = 1-cos^2 (x)
LHS = sin^5 (x) cos^2 (x) = sin x sin^4 (x) cos^2 (x) = sin x ( 1 - cos^2 (x) )^2 cos^2 (x) Just simplify and you would get the result.
the proof is sin^5(x)cos^2(x) = (cos^2(x) - 2cos^4(x) + cos^6(x))sin(x)
i'd like to say where i started but nothing i've tried has come even close so im missing an identity or common sense or something. any help or a shove in the right direction would be great. thanks so much.