Prosthaphaeresis Formulas

Theorem

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


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


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


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

Proof
These are proved in each case by expanding the RHS using the product-to-sum formulas:

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