$$ y = \left\{ \begin{array}{ll} \dfrac{\cos z}{z^2-\left( \dfrac{\pi}{2} \right)^2}, & z \ne \pm \dfrac{\pi}{2}\\ -\dfrac{1}{\pi}, & z = \pm \dfrac{\pi}{2}\\ \end{array} \right. $$
$$ \dfrac{\cos z}{\left(z-\dfrac{\pi}{2} \right)\left(z+\dfrac{\pi}{2}\right)}=\dfrac{\dfrac{e^{iz}+e^{-iz}}{2}}{\left(z-\dfrac{\pi}{2} \right)\left(z+\dfrac{\pi}{2}\right)}=\ldots $$
I need to prove that this is the entire function. I tried to write series representation, but I couldn't obtain as $\frac{\pi}2$.
Firstly, I need to obtain series representation and then I should show that it converges to $-\frac{1}{\pi}$ at $z=\frac{\pi}2$ , it should be analytic in all complex plane for be entire function.
How can I obtain that series?