# Sum to Product identity of Cos functions

## Formula

$\cos{\alpha}+\cos{\beta}$ $\,=\,$ $2\cos{\Big(\dfrac{\alpha+\beta}{2}\Big)}\cos{\Big(\dfrac{\alpha-\beta}{2}\Big)}$

The transformation of sum of two cosine functions into product form is called the sum to product identity of cos functions.

### Introduction

Take $\alpha$ and $\beta$ as angles of two right triangles. The cosines of them are written in mathematical form as $\cos{\alpha}$ and $\cos{\beta}$ in trigonometry. The addition of the two cosine functions are expressed in mathematical form as follows.

$\implies$ $\cos{\alpha}+\cos{\beta}$

In expressions and equations, the cosine functions are often involved in addition. So, the sum of the cos functions can be transformed into product form as follows.

$\implies$ $\cos{\alpha}+\cos{\beta}$ $\,=\,$ $2\cos{\Big(\dfrac{\alpha+\beta}{2}\Big)}\cos{\Big(\dfrac{\alpha-\beta}{2}\Big)}$

#### Popular forms

The rule of sum to product transformation of cosine functions is also popularly written in two other forms.

$(1) \,\,\,\,\,\,$ $\cos{x}+\cos{y}$ $\,=\,$ $2\cos{\Big(\dfrac{x+y}{2}\Big)}\cos{\Big(\dfrac{x-y}{2}\Big)}$

$(2) \,\,\,\,\,\,$ $\cos{C}+\cos{D}$ $\,=\,$ $2\cos{\Big(\dfrac{C+D}{2}\Big)}\cos{\Big(\dfrac{C-D}{2}\Big)}$

You can write the sum to product formula of cosine functions in terms of any two angles in the similar way.

#### Proof

Learn how to derive the sum to product transformation identity of cosine functions in trigonometry.

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