 Solution

Let $a=N+\frac{1}{2}$. The useful identity on the square contour isOn the vertical edges the first term equals one; on the horizontal edges $|y|=a$ and $\sinh \pi a>1$. Thus $|1/\sin \pi z|\le 1$ on every edge. Also $|z|\ge a$ and the contour has length $8a$, so the estimation lemma givesAt every nonzero integer $k$, the residue is $(-1{)}^k/(\pi k^2)$. At zero the Laurent series isso the residue there is $\pi /6$. The positive orientation and the residue theorem therefore giveLetting $N\to \infty$ yields the alternating reciprocal-square sum