# How to prove this identity π=∞∑k=−∞(sin(k)k)2\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;?

How to prove this identity?
I found the above interesting identity in the book $\bf \pi$ Unleashed.

Does anyone knows how to prove it?

Thanks.

Find a function whose Fourier coefficients are $\sin{k}/k$. Then evaluate the integral of the square of that function.

To wit, let

Then, if

then

where $a \in[0,\pi)$, using the function