Poisson summation formula clarification regarding Fejer kernel

How about trying the following alternative approach. First, note that for integer $ N $, the numerator of $ F_N(t) $ factors out since: $$ \sin^2(\pi N (t + n)) = \sin^2(\pi N t + \pi N n) = \sin^2(\pi N t) $$ The remaining part is given by the identity: $$ \sum_{n= -\infty}^{\infty} \frac{1}{\pi^2 (t+n)^2} = \frac{1}{\sin^2(\pi t)} $$ I am not entirely sure how to prove this identity...but I hope this will help anyway.