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 $\sin \theta \ne 0$.

Proof

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Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $161 \ \text{(b)}$