Sum to Product Identities

14 Aug 2022

These identities are rarely used but can be life-savers when they are applicable. There are four identities.

Here is the first identity.

\[\begin{align} \cos \alpha + \cos \beta &= \cos \left ( \frac{\alpha + \alpha}{2} \right ) + \cos \left ( \frac{\beta + \beta}{2} \right ) \\[10pt] &= \cos \left ( \frac{\alpha + \beta}{2} + \frac{\alpha - \beta}{2} \right ) + \cos \left ( \frac{\alpha + \beta}{2} - \frac{\alpha - \beta}{2} \right ) \\[10pt] &= \left [ \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) - \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] + \left [ \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) + \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] \\[10pt] &= 2 \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) \end{align}\]

Here is the second identity.

\[\begin{align} \cos \alpha - \cos \beta &= \cos \left ( \frac{\alpha + \alpha}{2} \right ) - \cos \left ( \frac{\beta + \beta}{2} \right ) \\[10pt] &= \cos \left ( \frac{\alpha + \beta}{2} + \frac{\alpha - \beta}{2} \right ) - \cos \left ( \frac{\alpha + \beta}{2} - \frac{\alpha - \beta}{2} \right ) \\[10pt] &= \left [ \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) - \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] - \left [ \cos \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) + \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] \\[10pt] &= 2 \sin \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \end{align}\]

Here is the third identity.

\[\begin{align} \sin \alpha + \sin \beta &= \sin \left ( \frac{\alpha + \alpha}{2} \right ) + \sin \left ( \frac{\beta + \beta}{2} \right ) \\[10pt] &= \sin \left ( \frac{\alpha + \beta}{2} + \frac{\alpha - \beta}{2} \right ) + \sin \left ( \frac{\alpha + \beta}{2} - \frac{\alpha - \beta}{2} \right ) \\[10pt] &= \left [ \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) + \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] + \left [ \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) - \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] \\[10pt] &= 2 \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) \end{align}\]

Here is the fourth identity.

\[\begin{align} \sin \alpha - \sin \beta &= \sin \left ( \frac{\alpha + \alpha}{2} \right ) - \sin \left ( \frac{\beta + \beta}{2} \right ) \\[10pt] &= \sin \left ( \frac{\alpha + \beta}{2} + \frac{\alpha - \beta}{2} \right ) - \sin \left ( \frac{\alpha + \beta}{2} - \frac{\alpha - \beta}{2} \right ) \\[10pt] &= \left [ \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) + \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] - \left [ \sin \left ( \frac{\alpha + \beta}{2} \right ) \cos \left ( \frac{\alpha - \beta}{2} \right ) - \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \right ] \\[10pt] &= 2 \cos \left ( \frac{\alpha + \beta}{2} \right ) \sin \left ( \frac{\alpha - \beta}{2} \right ) \end{align}\]