Sum-to-Product Formulas for Sine and Cosine

Theorem

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha + \sin \beta\)	\(=\)	\(\displaystyle 2 \sin \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha - \sin \beta\)	\(=\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \alpha + \cos \beta\)	\(=\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((4):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \alpha - \cos \beta\)	\(=\)	\(\displaystyle -2 \sin \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Proof

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

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle 2 \sin \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\sin \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right) + \sin \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Product-to-Sum Formulas for Sine and Cosine: $(3)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\sin \dfrac {2 \alpha} 2 + \sin \dfrac {2 \beta} 2} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \alpha + \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\sin \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right) - \sin \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Product-to-Sum Formulas for Sine and Cosine: $(4)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\sin \dfrac {2 \alpha} 2 - \sin \dfrac {2 \beta} 2} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \alpha - \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\((3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle 2 \cos \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\cos \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right) + \cos \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Product-to-Sum Formulas for Sine and Cosine: $(1)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 2 \frac {\cos \dfrac {2 \beta} 2 + \cos \dfrac {2 \alpha} 2} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \alpha + \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\((4):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle -2 \sin \left({\frac {\alpha + \beta} 2}\right) \sin \left({\frac {\alpha - \beta} 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -2 \frac {\cos \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right) - \cos \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Product-to-Sum Formulas for Sine and Cosine: $(2)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -2 \frac {\cos \dfrac {2 \beta} 2 - \cos \dfrac {2 \alpha} 2} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \alpha - \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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