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

### The Sum-to-Product Identities

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

Example:

Express $\mathrm{cos}\left(6x\right)+\mathrm{cos}\left(2x\right)$ as a product.

$\begin{array}{l}\mathrm{cos}\left(6x\right)+\mathrm{cos}\left(2x\right)=2\mathrm{cos}\left(\frac{6x+2x}{2}\right)\mathrm{cos}\left(\frac{6x-2x}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\mathrm{cos}\left(4x\right)\mathrm{cos}\left(2x\right)\end{array}$

### The Product-to-Sum Identities

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

Example:

Express the product $\mathrm{cos}\left(3x\right)\mathrm{sin}\left(2x\right)$ as a sum of trigonometric functions.

$\begin{array}{l}\mathrm{cos}\left(3x\right)\mathrm{sin}\left(2x\right)=\frac{1}{2}\left(\mathrm{sin}\left(3x+2x\right)-\mathrm{sin}\left(3x-2x\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}\left(\mathrm{sin}\left(5x\right)-\mathrm{sin}\left(x\right)\right)\end{array}$