Simpson's Formulas/Sine by Sine

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Theorem

$\sin \alpha \sin \beta = \dfrac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2$

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


Proof

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

$\blacksquare$


Sources