1 Trillion Followers Problem

$\large \begin{cases}{\sin(2x) + \sin(2y) = 1/3} \\ {\cos(2x) + \cos(2y) = 5/7}\end{cases}$

Given $x,y$ satisfy the system of equations above, and denote the value of $\tan(x) + \tan(y)$ as $\frac ab$ for coprime positive integers $a,b$.

Find the value of $a+b$.

Bonus: For the general system of equations below, state $\tan(x) + \tan(y)$ in terms of $A$ and $B$.

$\large \begin{cases}{\sin(2x) + \sin(2y) = A} \\ {\cos(2x) + \cos(2y) = B}\end{cases}$