Find $f(x)$ assuming that $f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$

If $f(x)$ is a real valued function such that $$f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$$Find $f(x)$.

I did $x\to\arcsin x$ and then $x\to \arccos x$ and I obtained $2\arcsin x=2\arccos x$ or $x=\frac{1}{\sqrt2}$

But I guess whatever I have found is absurd.

How to approach this question$?$ Any help is greatly appreciated.