Sine to Power of Odd Integer

Theorem

 * $\displaystyle \sin^{2n+1} \theta = \frac 1 {2^{2n}} \left({\sin \left({2n+1}\right) \theta - \binom{2n+1} 1 \sin \left({2n-1}\right) \theta + \cdots + \left({-1}\right)^n \binom{2n+1} n \sin \theta}\right)$

That is:
 * $\displaystyle \sin^{2n+1} \theta = \frac {\left({-1}\right)^n} {2^{2n}} \sum_{k \mathop = 0}^n \left({-1}\right)^k \binom {2n+1} k \sin \left({2n - 2k + 1}\right) \theta$