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Occasionally, we may meet some huge expressions, see $$ F(a)=\frac{1}{4}\sum_{n=1}^N \frac{\sin^2\big(\frac{(2n-1)\pi}{2N}\big)}{\Big[a^2-2a\cos \big(\frac{(2n-1)\pi}{2N}\big)+1\Big]^2} $$ where $a$ is a real number for instance. For simplicity we can assume that $\vert a\vert<1$. I just wonder that if a closed form exits or not, and what is it.

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My thoughts. To begin with, I try to find the generating function of $F(a)$, but it does not work at all. Afterwards, I find some identities in the book table of integrals series and products, which read enter image description here I hope that them will work. Any help?

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Considering let $z=ae^{it}$, we have

$$\cos t=\frac{z}{2a}+\frac{a}{2z}$$ $$\sin t=\frac{z}{2a}-\frac{a}{2z}$$

$$\frac{\sin t}{a^{2}-2a\cos t+1}= \frac{\frac{z}{2a}-\frac{a}{2z} }{|z-1|^{2}}= \frac{\frac{z}{a}-\frac{a}{z}}{2(z-1)\left( \frac{a}{z}-1 \right)}$$

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