The Sum-to-Product and Product-to-Sum Identities

The Sum-to-Product Identities

$\begin{array}{l} \sin (u) + \sin (v) = 2 \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) \\ \sin (u) - \sin (v) = 2 \cos (\frac{u + v}{2}) \sin (\frac{u - v}{2}) \\ \cos (u) + \cos (v) = 2 \cos (\frac{u + v}{2}) \cos (\frac{u - v}{2}) \\ \cos (u) - \cos (v) = - 2 \sin (\frac{u + v}{2}) \sin (\frac{u - v}{2}) \end{array}$

Example:

Express $\cos (6 x) + \cos (2 x)$ as a product.

$\begin{array}{l} \cos (6 x) + \cos (2 x) = 2 \cos (\frac{6 x + 2 x}{2}) \cos (\frac{6 x - 2 x}{2}) \\ = 2 \cos (4 x) \cos (2 x) \end{array}$

The Product-to-Sum Identities

$\begin{array}{l} \sin (u) \sin (v) = \frac{1}{2} [\cos (u - v) - \cos (u + v)] \\ \cos (u) \cos (v) = \frac{1}{2} [\cos (u - v) + \cos (u + v)] \\ \sin (u) \cos (v) = \frac{1}{2} [\sin (u + v) + \sin (u - v)] \end{array}$

Example:

Express the product $\cos (3 x) \sin (2 x)$ as a sum of trigonometric functions.

$\begin{array}{l} \cos (3 x) \sin (2 x) = \frac{1}{2} (\sin (3 x + 2 x) - \sin (3 x - 2 x)) \\ = \frac{1}{2} (\sin (5 x) - \sin (x)) \end{array}$

The Sum-to-Product and Product-to-Sum Identities

The Sum-to-Product Identities

The Product-to-Sum Identities

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