If $f$ is an even map $S^n$ to $S^n$ then this induces an map $S^n$ to $RP^n$ to $ S^n$ Also when n is odd we have that $H_n(RP^n)$ is isomorphic to $H_n(RP^n/RP^{n-1})$.
I would like to use this to prove that the degree of $f$ is even.
I do not know where to go from there.