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I want to show that $\sin(\frac{x}{2})(\sin(x)+\sin(2x)+...+\sin(nx))=\sin(\frac{nx}{2})\sin(\frac{n+1}{2}x)$

But I'm not quite sure how to start my proof. I tried to expand the left half of the equation but that didn't seem to be the right idea. Help appreciated!

Alex M.
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jordan1781
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  • For sum proofs like this, it's often worth beginning with mathematical induction. – πr8 Jan 10 '16 at 19:22
  • http://math.stackexchange.com/questions/1484286/prove-by-induction-sum-limits-k-1nsinkx-fracsin-fracn12xsin-f?rq=1, http://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro – Martin R Jan 10 '16 at 19:26

1 Answers1

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Hint:

Since $\sin x\sin y=\frac{1}{2}\cos(x-y)-\frac{1}{2}\cos (x+y)$ we have $$\sin \left(\frac{x}{2}\right)\sum_{k=1}^n\sin \left(kx\right) =\sum_{k=1}^n\left\{\tfrac12\cos \left[(k-\tfrac{1}{2})x\right]-\tfrac12\cos \left[(k+\tfrac12)x\right]\right\}$$ Which is telescopic.

Ángel Mario Gallegos
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