Where $a,b\in \Bbb{R}^n$, $|a|=5,|b|=10$, $a,b$ are linearly independent, define $\phi_a(x)$ to output the angular diameter of a point $x$ with respect to the sphere $S_1(a)=\{x\in \Bbb{R}^n:|x-a|=1\}$. The same with $\phi_b (x)$. Show that every local extreme point of $\phi_a(x)+\phi_b(x)$ on $S_1(0)$ is a linear combination of $a$ and $b$.
Hints: Show $\sin {{1\over 2}\phi_a(x)}={1\over |x-a|}$; use the gradient.
I have been spending hours trying to solve it. It is the first time I come across things such as multidimensional angles, angular diameters. No references were given in the course notes. Wikipedia provides a formula, with which showing the first hint is immediate, but I still can't tell if I am to use the formula, and if I am, I don't see how the form in the hint should assist me. With the formula I arrive at: $\phi_a(x)=2\arcsin({1\over |x-a|})$, $L=\phi_a(x)+\phi_b(x)-\lambda (|x|-1)$ and $\nabla L=(...,{-2{(x_i-a_i)\over |x-a|^2}\over (1+({1\over |x-a|})^2)^{1\over 2}}{-2{(x_i-b_i)\over |x-b|^2}\over (1+({1\over |x-b|})^2)^{1\over 2}}+\lambda x_i,...)$. It is monstrous and leads me nowhere. I also don't quite see what how one can use $a$ or $b$ length.