Triple Triple Angle Identity

Geometry Level 5

sin(9x)3sin(3x)=(1sin2(x)sin2(p))(1sin2(x)sin2(q))(1sin2(x)sin2(r)) \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,qp,q and rr such that 0p,q,rπ2 0 \leq p,q,r\leq \frac \pi2 . If p+q+r=ABπp + q + r = \dfrac AB \pi for coprime positive integers AA and BB, find the value of A+BA+B.

Bonus: Generalize the trigonometric identity.

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