A trig identity

Geometry Level 4

Given that \( \cos2\theta = \dfrac{1+\sqrt5}4 \), if \( \tan^2 (3\theta) \tan^2(\theta) \) is equal to \( \dfrac mn\), where \(m\) and \(n\) are coprime positive integers, find \(m+n\).


Adapted from Purdue Problem of the week.
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