# Cosine angle difference identity

## Formula

$(1).\,\,$ $\cos{(a-b)}$ $\,=\,$ $\cos{a}\cos{b}$ $+$ $\sin{a}\sin{b}$

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

### Introduction

Let $a$ and $b$ be two variables, which are used to represent two angles in this case. The subtraction of angle $b$ from angle $a$ is the difference between them, and it is written as $a-b$, which is a compound angle. The cosine of a compound angle $a$ minus $b$ is written as $\cos{(a-b)}$ in trigonometry. The cosine of subtraction of angle $b$ from angle $a$ is equal to the sum of the products of cosines of angles $a$ and $b$, and sines of angles $a$ and $b$.

$\cos{(a-b)}$ $\,=\,$ $\cos{a} \times \cos{b}$ $+$ $\sin{a} \times \sin{b}$

This mathematical equation can be called the cosine angle difference trigonometric identity in mathematics.

#### Usage

The cosine angle difference identity is possibly used in two different cases in trigonometry.

##### Expansion

The cosine of difference of two angles is expanded as the sum of the products of cosines of angles and sines of angles.

$\implies$ $\cos{(a-b)}$ $\,=\,$ $\cos{(a)}\cos{(b)}$ $+$ $\sin{(a)}\sin{(b)}$

##### Simplification

The sum of the products of cosines of angles and sines of angles is simplified as the cosine of difference of two angles.

$\implies$ $\cos{(a)}\cos{(b)}$ $+$ $\sin{(a)}\sin{(b)}$ $\,=\,$ $\cos{(a-b)}$

#### Forms

The angle difference identity in cosine function is written in several forms but the following three forms are some popularly used forms in the world.

$(1).\,\,$ $\cos{(A-B)}$ $\,=\,$ $\cos{A}\cos{B}$ $+$ $\sin{A}\sin{B}$

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

$(3).\,\,$ $\cos{(\alpha-\beta)}$ $\,=\,$ $\cos{\alpha}\cos{\beta}$ $+$ $\sin{\alpha}\sin{\beta}$

#### Proof

Learn how to derive the cosine of angle difference trigonometric identity by a geometric method in trigonometry.

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