Too Complex! Part- III

Geometry Level 4

$\sin 5\theta = 5 \sin \theta \left ( 1 - \frac{\sin^{2} \theta}{\sin ^{2} p} \right) \left( 1 - \frac{\sin^{2} \theta}{\sin ^{2} q} \right)$

If the equation above is a trigonometric identities for constants $$p$$ and $$q$$ such that $$0 \leq p,q\leq \frac \pi2$$, find the value of $$p + q$$.

Clue: Use the equation $$z^{10} - 1 = 0$$

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