Show that the taylor series of $\cos{\frac{\pi}{2}}$ equals $\cos{\frac{\pi}{2}}$

Question

Show that the taylor series of $\cos{\frac{\pi}{2}}$ equals $\cos{\frac{\pi}{2}}$.

I have done the taylor series as follows:

$\cos{\frac{\pi}{2}}=\sum_{n=2k+1}^{\infty}(-1)^k\frac{\left(x-\frac{\pi}{2}\right)^n}{n!}$

I have two questions:

• Is the formula right?

• how can I show that this sum is equals really $\cos{\frac{\pi}{2}}$?