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I am stumped by the sum $$\sum_{x=0}^n \binom{n}{x}\sin\big(\frac{\pi x}{n}\big)$$ but I can't figure it out. I tried expanding the taylor series of sine and using Euler's identity, but to no avail. Any hints?

PLEASE do not give me a full solution - I just need a hint.

Thanks!

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    $\begingroup$ Hint: $\sin \theta = \mathrm{Im} \: e^{i\theta}$. Forgive me if this is too big of spoiler! But, well, how could I have said any less? $\endgroup$
    – Hamed
    Commented Jul 25, 2017 at 23:44
  • $\begingroup$ (+1) for the last line $\endgroup$
    – Claude Leibovici
    Commented Jun 13 at 5:04

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