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.