The sines speak otherwise

Geometry Level 4

Given that \(\dfrac{\sin x}{\sin y} = 3 \) and \(\dfrac{\cos x}{\cos y} = \dfrac{1}{2} \).

The value of \(\dfrac{\sin2x}{\sin2y}- \dfrac{\cos2x}{\cos2y} \) can be expressed as \( \dfrac{p}{q} \) for coprime positive integers \(p\) and \(q\). Find \(p+q\).


This is a modified AIME problem.
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