1 Trillion Followers Problem

Geometry Level 5

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

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 \(B\).

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

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