How to prove $\gcd(k^{(2^i)}+1,k^{(2^j)}+1)=1$?

Question

I want to know how to prove the following formula

1)$\gcd(k^{2^i}+1,k^{2^j}+1)=1$ when $i \ne j$ and $k$ is even.

2)$\gcd(k^{2^i}+1,k^{2^j}+1)=2$ when $i \ne j$ and k is odd.

Thanks!