Finding the exact sum of this power series

Question

I am currently studying power series and have come across a problem I am having difficulties with. I have done some looking around on the website for a similar problem but I cant find anything that doesn't employ the lim sup method, which I am unable to use yet. Any help/guidance would be greatly appreciated!

The problem is to exactly sum the following power series:

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

Thank you.

Answer 1

Hint: $$\cos x=\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}$$

Answer 2

Tip: Consider $e^{1+i\pi}+e^{1-i\pi}-2$.