Prove that $f(\alpha)=\sin^{-1}(n\sin(\alpha))+\sin^{-1}(n\sin(\varphi-\alpha))-\varphi$ has an extrema at $\alpha=\frac{\varphi}{2}$, with $0 \leq \alpha\leq \varphi \leq \frac{\pi}{2}$ and $n \sin(\varphi) \leq1$
It comes from a geometric optics problem: $f(\alpha)$ is the angle of deviation of a prism.
Of course, I can do it with differentiation, but that's a bit messy, and I'd like to know if there's an elegant solution using some kind of symmetry.