<s>[INST] Find the minimum of $\displaystyle \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$ Is it possible to find the minimum value of $E$ where
$$E = \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$$for any $\triangle ABC$.
I've got the feeling that $\min(E) = 4$ and that the critical value occurs when $ABC$ is equilateral.
 [/INST] HINT: Use Lagrange multipliers to find the minimum of the function $f(A,B,C) = \frac{1}{\sin^2(A)} + \frac{1}{\sin^2(B)} +\frac{1}{\sin^2(C)}$ with the constraint $A+B+C = \pi$ 
</s>