 Solution

For an isolated singularity at $a$, write the Laurent expansion in a punctured disk as $f(z)={\sum }_{k=-\infty }^{\infty }{c}_k(z-a{)}^k$. Its residue isThe sine in the denominator has simple zeros precisely at the integers, and the numerator is nonzero there. Consequently $g(z)=\pi \cot (\pi z)$ has simple poles exactly at $n\in \mathbb{Z}$, withThere are no other finite singularities.