Basic formula manipulation related with Legendre symbol

Question

Using the fact that $i$ = cos$\frac \pi2$ + $i$sin$\frac \pi2$, show that $$\frac{(1+i^p)i^{-p/2}}{(1+i)i^{-1/2}} = \frac{cos\frac {p\pi}4}{cos\frac \pi4}$$

I found this problem when studying about Legendre symbol so I tag it as number theory.

Answer 1

Hint:

$$(1+i^{2k})i^{-k}=i^k+i^{-k}=2\cos\frac{k\pi}{2}$$

as $i^k= \cos\frac{k\pi}{2}+i\sin\frac{k\pi}{2}$ and $i^{-k}=?$