Simpson's Formulas

Theorem

 * $\displaystyle (1) \qquad \cos \alpha \cos \beta = \frac {\cos \left({\alpha - \beta}\right) + \cos \left({\alpha + \beta}\right)} 2$


 * $\displaystyle (2) \qquad \sin \alpha \sin \beta = \frac {\cos \left({\alpha - \beta}\right) - \cos \left({\alpha + \beta}\right)} 2$


 * $\displaystyle (3) \qquad \sin \alpha \cos \beta = \frac {\sin \left({\alpha + \beta}\right) + \sin \left({\alpha - \beta}\right)} 2$


 * $\displaystyle (4) \qquad \cos \alpha \sin \beta = \frac {\sin \left({\alpha + \beta}\right) - \sin \left({\alpha - \beta}\right)} 2$

Proof
These are proved in each case by expanding the RHS using the Sine and Cosine of Sum formulas:

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