Product-to-Sum Formulas for Sine and Cosine

Theorem

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

Proof

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

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \frac {\cos \left({\alpha - \beta}\right) + \cos \left({\alpha + \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\left({\cos \alpha \cos \beta + \sin \alpha \sin \beta}\right) + \left({\cos \alpha \cos \beta - \sin \alpha \sin \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {2 \cos \alpha \cos \beta} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \alpha \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

\((3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \frac {\sin \left({\alpha + \beta}\right) + \sin \left({\alpha - \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\left({\sin \alpha \cos \beta + \cos \alpha \sin \beta}\right) + \left({\sin \alpha \cos \beta - \cos \alpha \sin \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {2 \sin \alpha \cos \beta} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \alpha \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\((4):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \frac {\sin \left({\alpha + \beta}\right) - \sin \left({\alpha - \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\left({\sin \alpha \cos \beta + \cos \alpha \sin \beta}\right) - \left({\sin \alpha \cos \beta - \cos \alpha \sin \beta}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {2 \cos \alpha \sin \beta} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \alpha \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$