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.