The sines speak otherwise...

Geometry Level 4

Given that \( \frac{\sin x}{\sin y} = 3 \) and \( \frac{\cos x}{\cos y} = \frac{1}{2} \), the value of \( \frac{\sin2x}{\sin2y}- \frac{\cos2x}{\cos2y} \) can be expressed as \( \frac{p}{q} \) for positive coprime integers \(p\) and \(q\). Find \(p+q\).

Note: This is a modified AIME problem.

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