[Math] Prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$

Question

I need to prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$. I have already checked that this is correct for $n=1$ and $n=2$, but I'm not able to prove this identity by induction. Now I was thinking of making a shift to the imaginary numbers by saying : $\sin(nx) + \sin((n+2) x) = Im\{e^{i n x}(1 + e^{2ix})$, but I have no clue how to continue.

Answer 1

Recall the formula $$\sin(A) + \sin(B) = 2 \cos((B-A)/2) \sin((A+B)/2)$$