Quintuple Angle Formulas/Sine/Corollary

Theorem
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
 * $\dfrac {\map \sin {5 \theta} } {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$

where $\sin$ denotes sine and $\cos$ denotes cosine.

Proof
First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$:
 * $\sin \theta = 0$

so $\dfrac {\map \sin {5 \theta} } {\sin \theta}$ is undefined.

Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$