If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$.
I don't know how to solve it. And can you please mention the used formulas first.
What I can see is that if one of the angles $A$, $B$, $C$ is small, then the value $\cot^2A$ or $\cot^2B$ or $\cot^2C$ will be big. So I want to make angles big (more precisely, close to $\pi/2$, where cotangent is zero), but the condition $A+B+C=\pi$ prevents me from making all three of them very big .
I can see that if $A=B=C=\frac\pi3$, then I get $\cot A=\cot B=\cot C=\frac1{\sqrt3}$ and $\cot^2A+\cot^2B+\cot^2C=1$. But I do not know whether this is indeed minimum. (According to WolframAlpha this is the minimum. However, I would like to see some proof of this fact.)
Best Answer
For another way, since $\cot^2x$ is convex when $x \in [0,\pi]$ and we need to worry only of positive $x$, so by Jensen's inequality: $$ \sum_{cyc} \cot^2 A \geqslant 3 \cot^2\frac{A+B+C}3=1$$
As equality is possible when $A=B=C=\frac{\pi}3$, we have the minimum.