Simpson's Formulas/Sine by Sine

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Theorem

$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$

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


Proof

\(\displaystyle \) \(\) \(\displaystyle \frac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} - \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2\) Cosine of Difference and Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 \sin \alpha \sin \beta} 2\)
\(\displaystyle \) \(=\) \(\displaystyle \sin \alpha \sin \beta\)

$\blacksquare$


Sources