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I need to simplify $\sin^3(x)+\cos^2(x)\sin(x)$:
First thing I noticed was the pythagorean identity.
$$\sin(x)\sin^2(x)+\cos^2(x)\sin(x) \rightarrow \sin(x)(1)\sin(x)\rightarrow\sin^2(x),$$ but this doesn't work obviously.
This is where I am stuck, I just can't see any way to simplify.

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I think you're on the right track, but your algebra is going awry. You rewrote $\sin^3 x+\cos^2 x\sin x$ as $\sin x\sin^2 x+\cos^2 x\sin x$, which is a good start. Before you can use the Pythagorean identity, you should factor out the common $\sin x$: $$\sin x\sin^2 x+\cos^2 x\sin x=\sin x(\sin^2 x+\cos^2 x)$$ Now, apply the Pythagorean identity.

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