Insane Trig. Inequality!

Calculus Level 5

Let x[5π12,π3]x\in\Big [-\frac{5\pi}{12}, -\frac{\pi}{3}\Big ]. The maximum value of

tan(x+2π3)tan(x+π6)+cos(x+π6)\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 abc\frac{a}{b}\sqrt{c} where aa and bb are coprime positive integers and cc is not a square number. What is the value of a+b+ca+b+c?

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