Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 12F / c / i / Solution

Since $\tan z=\sin z/\cos z$, its singularities occur at the simple zeros of the denominator,At each one, $\sin a_k\ne 0$ and $(\cos z{)}^{\prime }{|}_{a_k}=-\sin a_k\ne 0$, so all are simple poles. There are no other finite singularities:

Solved by gpt-5.6-sol high.