what are the poles of $\cot z$?

Question

I have this in my textbook.

The function $f_{2}(z)=\cot (z)=\frac{\cos z}{\sin z}$ has a simple pole at $z=0 .$ The residue is $ \operatorname{Res}\left(f_{2}(z), z=0\right)=\lim _{z \rightarrow 0} \frac{z \cos z}{\sin z}=1 $

But why $\cot z$ has a simple pole at $0$? and no other poles elsewhere? thanks

Answer 1

It has a simple pole at $0$ because $\lim_{z\to0}z\cot z$ exists (in $\mathbb C$) and it is different from $0$. But it also has a simple pole at every multiple integer of $\pi$.