1 Trillion Followers Problem

Geometry Level 5

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

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

Find the value of a+ba+b.

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

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

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