# Sum to Product identity of Cosine functions

## Formula

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

An identity that expresses the transformation of sum of cosine functions into product form is called the sum to product identity of cosine functions.

### Introduction

When $\alpha$ and $\beta$ represent two angles of the right triangles. The cosine of the two angles are written as $\cos{\alpha}$ and $\cos{\beta}$ in trigonometric mathematics. The sum of the two cosine functions is written mathematically as follows.

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

The sum of cosine functions can be transformed into the product of the trigonometric functions in the following mathematical form.

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

#### Other forms

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

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

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

Thus, you can write the sum to product transformation formula for cosine functions in terms of any two angles.

#### Proof

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

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Jun 26, 2023

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