Since the integrand decays like $1/|z|^2$
$$
\oint\pi\cot(\pi z){\small\left(\frac1{(8z+1)^2}-\frac1{(8z+3)^2}-\frac1{(8z+5)^2}+\frac1{(8z+7)^2}\right)}\,\mathrm{d}z=0\tag{1}
$$
There are singularities at each integer and at $\left\{-\frac18,-\frac38,-\frac58,-\frac78\right\}$.

The residue of $\pi\cot(\pi z)$ is $1$ at each integer, so the contribution of the residues at the integers is
$$
2\sum_{k=0}^\infty\left(\frac1{(8k+1)^2}-\frac1{(8k+3)^2}-\frac1{(8k+5)^2}+\frac1{(8k+7)^2}\right)\tag{2}

Mathematician Love letter #1

My Dear Omega,

Yesterday, I was passing by your rectangular house in face,conical nose and spherical eyes,standing in your triangular garden. Before seeing you my heart was a null set, but when a vector of magnitude (likeness) from your eyes at a deviation of theta radians made a tangent to my heart, it differentiated. My love for you is a quadratic equation with real roots, which only you can solve by making good binary relation with me.

The cosine of my love for you extends to infinity.

Mathematician Love Letter #2

Dear Sigma,

With reference to the syllabus of my feelings, I want to prove that the locus of my point is directly proportional to your heart. On seeing you, I feel like a triangle with only two angles; the third being you! At times it seems to me that I am a circle without a radius or a matrix without determinants. We are like two simultaneous equations without any solutions.

You never seem to notice that I am a point lying on your vector equation, or in other words, we are collinear. Your smile makes my head rotates 360 degrees anticlockwise and on applying …