# Difference to Product identity of Cos functions

## Formula

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

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

### Introduction

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

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

The cos functions are often involved in subtraction in some expressions and equations. So, the difference of the cos functions can be transformed into product form as follows.

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

#### Popular forms

The difference to product transformation rule of cosine functions is also popularly written in two other 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, you can write the difference to product formula of cosine functions in terms of any two angles.

#### Proof

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

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