System of equations

Geometry Level 4

Given that \(\dfrac{\sin x}{\sin y}=4\) and \(\dfrac{\cos x}{\cos y}=\dfrac{1}{3}\).

And if \(\dfrac{\sin 2x}{\sin 2y}+\dfrac{\cos 2x}{\cos 2y}=\dfrac{m}{n}\), where \(m,n\) are co-prime positive integers, find the value of \(m+n\).

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