Odd Multiple Angle Formula for Sine

Theorem

 * $\ds \frac {\map \sin {2 n + 1} \theta} {\sin \theta} = \paren {2 n + 1} \prod_{k \mathop = 1}^n \paren {1 - \frac {\sin^2 \theta} {\map {\sin^2} {\frac {k \pi} {2 n + 1} } } }$

for $\sin \theta \ne 0$.