Series representaion of $\frac {\pi}{\sin{\pi z}}$?

Question

Suppose i know the following two formulas:

$\displaystyle \sin(\pi z) = \pi z \prod_{n=1}^\infty \left( 1 - \frac{z^2}{n^2} \right)$ and $\pi \cot(\pi z)= \sum_{n=-\infty}^{n=\infty} \frac {1}{z+n}$ Then i am trying to prove that

$\frac {\pi}{\sin{\pi z}}=\sum_{n=-\infty}^{n=\infty} \frac {(-1)^n}{z-n}$

I am not getting the idea for this,any hints/ideas?

Answer 1

Hint. Use Hadamard Factorization Theorem.