The sines speak otherwise

Geometry Level 4

Given that sinxsiny=3\dfrac{\sin x}{\sin y} = 3 and cosxcosy=12\dfrac{\cos x}{\cos y} = \dfrac{1}{2} .

The value of sin2xsin2ycos2xcos2y\dfrac{\sin2x}{\sin2y}- \dfrac{\cos2x}{\cos2y} can be expressed as pq \dfrac{p}{q} for coprime positive integers pp and qq. Find p+qp+q.


This is a modified AIME problem.
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