# Insane Trig. Inequality!

Calculus Level 5

Let $x\in\Big [-\frac{5\pi}{12}, -\frac{\pi}{3}\Big ]$. The maximum value of

$\tan\bigg (x+\frac{2\pi}{3}\bigg )-\tan\bigg (x+\frac{\pi}{6} \bigg )+\cos\bigg (x+\frac{\pi}{6}\bigg )$

can be expressed as $\frac{a}{b}\sqrt{c}$ where $a$ and $b$ are coprime positive integers and $c$ is not a square number. What is the value of $a+b+c$?

×