TITLE: Is there a function satisfying the following equation: $f(\sin x)+f(\cos x) = \frac{\tan x}{2}$?
QUESTION [9 upvotes]: Define a function $f(x)$ such that:
$$f(\sin x)+f(\cos x)=\frac{\tan x}2$$
What is $f(x)?$ 
My attempt: I hypothesized the denominator of the function to be like of the form $x+(1-x^2)^{1/2}$

REPLY [26 votes]: More general than gimusi's answer, notice that
$$\sin(x)=\cos(\pi/2-x)\\\cos(x)=\sin(\pi/2-x)$$
And so
$$f(\sin(x))+f(\cos(x))=f(\sin(\pi/2-x))+f(\cos(\pi/2-x))$$
But
$$\tan(x)\ne\tan(\pi/2-x)$$