Math Doubts

Sum to Product identity of Cos functions

Formula

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

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

Introduction

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

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

In expressions and equations, the cosine functions are often involved in addition. So, the sum of the cos functions can be transformed into product form as follows.

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

Popular forms

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

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

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

You can write the sum to product formula of cosine functions in terms of any two angles in the similar way.

Proof

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

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