How can I prove that it is an Entire Function

Question

Prove that if

$$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$

then $f$ is an entire function.

Answer 1

HINTS:

$$\frac{1}{z^2-(\pi/2)^2}=\frac1\pi \left(\frac{1}{z-\pi/2}-\frac{1}{z+\pi/2}\right)$$

and

$$\cos(z)=\sum_{n=1}^\infty (-1)^n\frac{(z-\pi/2)^{2n-1}}{(2n-1)!}$$

and

$$\cos(z)=\sum_{n=1}^\infty (-1)^{n-1}\frac{(z+\pi/2)^{2n-1}}{(2n-1)!}$$