Product and Sum Identities

Similar to the Double Angle formulae, generally these equations are only used for simplification and manipulation of equations.

$\cos \theta \cos \phi = \frac{\cos(\theta - \phi) + \cos(\theta + \phi)}{2}$

$\sin \theta \sin \phi = \frac{\cos(\theta - \phi) - \cos(\theta + \phi)}{2}$

$\sin \theta \cos \phi = \frac{\sin(\theta + \phi) + \sin(\theta - \phi)}{2}$

$\cos \theta \sin \phi = \frac{\sin(\theta + \phi) - \sin(\theta - \phi)}{2}$

The sum identities are similar, but allow you to go the other way:

$\cos \theta + \cos \phi = 2 \cos(\frac{\theta + \phi}{2}) \cos(\frac{\theta - \phi}{2})$

$\cos \theta - \cos \phi = -2 \sin(\frac{\theta + \phi}{2}) \sin(\frac{\theta - \phi}{2})$

The final sum identity for sin uses the ± notation. This just allows us to write the identity once for both sin + sin, and sin – sin. (Note that the reversal of the ± inside the second cosine just indicates that we use the opposite sign to the others.)

$\sin \theta \pm \sin \phi = 2 \sin(\frac{\theta \pm \phi}{2}) \cos(\frac{\theta \mp \phi}{2})$

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