So:

f(-x)+g(-x)=tan(-x+π/4)

The function f(x) is odd function and the function g(x) is even.

Namely: f(-x)=-f(x), g(-x)=g(x)

So there are:

-f(x)+g(x)=tan(-x+π/4).......(2)

( 1)+(2):

g(x)=[tan(x+π/4)+tan(-x+π/4)]/2

( 1)-(2):

f(x)=[tan(x+π/4)-tan(-x+π/4)]/2

You just need to simplify it again.