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 \(a+b+c\)?

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