<s>[INST] Show that $\sin(3x)$ is equivalent to $3\sin(x)\cos^2(x)-\sin^3(x)$ 
The expression $\sin(3x)$ is equivalent to:
A. ...

My book states the right answer is B which is $3\sin(x)\cos^2(x)-\sin^3(x)$.
I tried:
$$\sin(x)\cos(2x)+\cos(x)\sin(2x) = \\
\sin(x)(2\cos^2(x)-1)+\cos(x)\cdot2\sin(x)\cos(x) = \\
2\cdot\sin(x)\cos(x)\cos(x)-\sin(x)+\cos(x)2\sin(x)\cos(x) = \\
\sin(x)(2\cos^2(x)-1+2\cos^2(x)) = \\
\sin(x)(4\cos^2(x)-1) = \\
4\cos^2(x)\sin(x)-\sin(x) = ???$$
What do I do next?
 [/INST] $$\sin(3x) = \sin(x+2x)$$
$$= \sin(x) \cos(2x) + \sin(2x) \cos(x)$$
$$ = \sin(x) ( \cos^2(x) - \sin^2 (x)) + \cos(x) (2 \sin(x) \cos(x)) $$
$$= 3 \sin(x) \cos^2 (x) - \sin^3 (x )$$
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