Show that $f(z)=\tan(z)$ is analytic in $\mathbb{C}$ excpet for simple poles at $z=(n+\frac{1}{2})\pi$ for $n \in \mathbb{Z}$. Determine the singular part of $f$ at these poles.
My thought is to consider the power series for both $\sin(z)$ and $\cos(z)$ and since those are entire deal with what happens at the poles. I'm not really sure about the singular part here because I have two general power series and there doesn't seem to be a good way of dealing with it.