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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