Function inverse based on restricted domain,; proof for max/min value of the function

Question

A continuous function $f$ is such that $\frac{f(x+y)}{f(x)+f(y)}=\frac{f(x)-f(y)}{f(x-y)}.$ Additionally, the function is periodic with period of $2\pi.$ Finally, the function is question has no value of $k$ such that for $f(x+k)=f(x)$ for any real number $x$ and value of $k<2\pi.$

Prove that $f(\frac{\pi}{2}+2\pi n)$ are either the maximums or minimums of the function.

I am assuming that the last given constraint of the function means that the function has an inverse with a restricted domain of $[0,2\pi),$ but am not sure how this can be a continuous function.

Answer 1

Letting x = y in the equation, one gets f(x + x) = 0.
Whence f is the constant 0 function which contradicts the condition
f(x + k) /= f(x) for k < 2$\pi$.
A condition impossible to meet because
f(x + 0) = f(x) and 0 < 2$\pi$.
However, for all x, f(x) is both maximal and minimal.
Thusly the problem self-destructs.