# Difference to Product identity of Cosine functions

## Formula

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

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

### Introduction

If $\alpha$ and $\beta$ represent the two angles of right triangles, then the cosine functions with the two angles are written in mathematical form as $\cos{\alpha}$ and $\cos{\beta}$. The difference of the cosine functions is written mathematically in the following mathematical form.

$\cos{\alpha}-\cos{\beta}$

The difference of cosine functions can be converted into the product of trigonometric functions as follows.

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

#### Other forms

The difference to product transformation for the cosine functions is written in two popular forms.

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

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

Thus, we can write the difference to product transformation formula for cosine functions in terms of any two different angles.

#### Proof

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

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