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.