Triple Triple Angle Identity

Geometry Level 5

\[ \dfrac{\sin(9x)}{3\sin(3x)} = \left(1 - \frac{\sin^2(x)}{\sin^2(p)}\right)\left(1 - \frac{\sin^2(x)}{\sin^2(q)}\right)\left(1 - \frac{\sin^2(x)}{\sin^2(r)}\right) \]

Above shows a trigonometric identity with constants \(p,q \) and \(r\) such that \( 0 \leq p,q,r\leq \frac \pi2 \). If \(p + q + r = \dfrac AB \pi\) for coprime positive integers \(A\) and \(B\), find the value of \(A+B\).

Bonus: Generalize the trigonometric identity.

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