<s>[INST] Proving the identity: $\sin3x + \sin x = 2\sin2x\cos x$ I need some help proving this identity: 
$$\sin3x  + \sin x = 2\sin2x\cos x$$
I don't know where to start. I thought about expanding $\sin 3x$ into $\sin (2x + x)$ but I don't think that does me any good. Any hints would be appreciated.
Thanks!
 [/INST] $$\sin 3x+\sin x=2\sin 2x\cos x$$
$$LHS
=\sin(2x+x)+\sin x\\
=\sin2xcosx+\sin x\cos2x+\sin x\\
=2(\sin x \cos x)\cos x+\sin(cos^2x-\sin^2x)+\sin x\\
=2\sin x \cos^2x+\sin x \cos^2x-\sin^3x+\sin x\\
=\cos^2x(2\sin x+\sin x)-\sin^3x+\sin x\\ 
=(1-\sin^2x)3\sin x-\sin^3x+\sin x\\
=3\sin x-\sin^3x-\sin^3x+\sin x\\
=4\sin x-4\sin^3x\\
=4(\sin x-\sin x(1-\cos^2x)\\
=4(\sin x \cos^2x)\\
=4\sin x \cos^2x$$
RHS
$$=2(2\sin x \cos x)\cos x\\
=4\sin x \cos^2x$$
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