Prove that $\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \le 3(\sqrt{3}-1)$

What about Calculus:

$f(x)=\frac{1}{1+\tan(x)}$ has a positive second derivative on $(\frac{\pi}{4}, \frac{\pi}{2})$. This solves the problem.