Inspired by Shivamani Patil

Geometry Level 4

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

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