I was hoping for a proof of something which appears to be intuitive to me, but which I can't prove.
Let $a, b$ & $c$ be lengths of the sides of a triangle. We know that on a plane, $c^2 = a^2 + b^2$ when $C$ is a right angle.
Does a similar identity $c^2 \leq a^2 + b^2$ hold on a sphere?
On a sphere, when $a$, $b$ or both are $0$, then $c^2=a^2 + b^2$. For practical case $a>0$ and $b>0$, and $C$ being a right angle , $c^2$ has always been less than $a^2 + b^2$ for whatever examples I have tried.
I have read the Pythagorean theorem, the sine rule, the cosine rule, and other basic identities for spherical triangles, but I have not been able to prove this particular identity.
In case the identity what I'm trying to prove does not hold, would appreciate a counter example.
Any help is appreciated!