Prove $\sin((2n+1)x)$ function by induction

Question

Can someone help me prove the following by mathematical induction:

$$\sin((2n+1)x)=\sin(x)(1+2 \sum_{k=1}^{n} \cos(2kx))$$

I was told to use induction on $n$; however I keep getting stuck. Any help would be greatly appreciated!

Answer 1

If $n=0$, this is trivially true. Suppose $n-1$ is true, let's consider $n$, it suffices to show $$\sin((2n+1)x)-\sin((2n-1)x)=2\sin(x)\cos(2nx)$$ now do you see how to finish?