 Solution

Use the same normalized spherical cap area $q$, constant $c_n$ and function $F={\sin }^n\rho -q$ as above. Put ${\rho }_0=\arcsin (1/\sqrt{n})$. The supplied ratio estimate givesThe middle inequality holds for every $n\ge 2$, since $n+1\le 2\pi (n-1)$. Applying the upper projection bound for the spherical cap at ${\rho }_0$ yieldsso $F({\rho }_0)\ge 0$.

RecallAt ${\rho }_0$ the expression in parentheses is positive, because $n\sin {\rho }_0\cos {\rho }_0=\sqrt{n-1}$ and $(n-1)c_n<\sqrt{n-1}$. It increases up to $\pi /4$ and then decreases strictly to a negative value at $\pi /2$. Hence it crosses zero exactly once on $[{\rho }_0,\pi /2]$, from positive to negative. The minimum of $F$ is again an endpoint value; $F({\rho }_0)\ge 0$ and $F(\pi /2)=1/2$. This proves the stronger range of the spherical cap area upper bound: