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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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    @Norbert - $cis(\theta):=\cos(\theta)+i\sin(\theta)$2012-10-31

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Hint:

  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) $

  2. Recall geometric series formula.