Want to prove certain sum representation of $\cot(x)$

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So here is my problem,

I would like to prove an identity I found in a book which was given without a proof.

Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I thought since there is no explicit proof it would be easy to show it but I failed...

Could someone help me? Or does anybody know where I can find the proof?

Thank you!