how to construct this function using identities

Question

Given that $f(\sin x)+f(\cos x)= \tan x$.

My attempt to find $f(x)$ is by guessing terms to meet the condition. Please help.

Note $0 < x < \frac{\pi}{2}$.

Answer 1

So looking at $x = \pi/6$, we have

\begin{align} f(\sin \pi/6) + f (\cos \pi/6) &= \tan \pi/6 \\ f(0.5) + f(0.866...) &\approx .577, \end{align} but for $x = \pi/3$, we get \begin{align} f(\sin \pi/3) + f (\cos \pi/3) &= \tan \pi/3 \\ f(0.866...) + f(0.5) &\approx 1.73. \end{align}

That just can't happen, because addition is commutative.