I am thinking about the Laurent series of the function:

$$f(z)=\frac{1}{\sin z}$$

How could I define its annulus of convergence? Since the singularities are $$z=m\pi, m\in\mathbb{Z}$$

Aren't there infinitely many annulus' of convergence for which I can write a Laurent series. For example, one could be:

$$C=\{z\in\mathbb{C}\ |\ \pi<\left|z\right|<2\pi\}$$

So, is it possible to have many different annulus' of convergence? Also, could
$$C=\{z\in\mathbb{C}\ |\ 0<\left|z\right|<\pi\}$$ 
be an annulus of convergence?