Inspired by Shivamani Patil

Geometry Level 3

$\dfrac{\tan(x + 120^{\circ})}{\tan(x - 30^{\circ})} = \dfrac{11}{2}$

If $x$ is a solution to the above equation and $\cos(4x) = \dfrac{a}{b},$ where $a$ and $b$ are coprime positive integers, then find $a + b.$