Exponentiating Sine

Algebra Level 3

Suppose \(\theta\) is an angle strictly between \(0\) and \(\frac{\pi}{2}\) such that \(\sin 5\theta = \sin ^5 \theta \). The number \(\tan 2\theta \) can be uniquely written as \(a \sqrt{b}\), where \(a\) and \(b\) are positive integers, and \(b\) is not divisible by the square of a prime. What is the value of \(a+b\)?

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