# How to calculate $\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$?

I was given an exercise:

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

I recognize $$\sin(kx)=Im(cis(kx))=Im(cis^{k}(x))$$ and $$\sin^{k}(x)=(Im(cis(x)))^{k}$$ but I do not know how to proceed .

I would appreciate any help or hint on how to get started, I guess that is should be related to geometric series, but I didn't manage to get to any geometric series.

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What is $cis$? ${}{}$ –  no identity Oct 31 '12 at 14:15
@Norbert - $cis(\theta):=\cos(\theta)+i\sin(\theta)$ –  Belgi Oct 31 '12 at 14:16

1. Prove that $$\frac{\sin(kx)}{\sin^k(x)}=\frac{1}{2i}\left(\left(\frac{e^{i x}}{\sin x}\right)^k-\left(\frac{e^{-i x}}{\sin x}\right)^k\right)$$