# Cosine angle sum 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 us consider that $a$ and $b$ are two variables, which denote two angles. The sum of two angles is written as $a+b$, which is actually a compound angle. The cosine of a compound angle $a$ plus $b$ is expressed as $\cos{(a+b)}$ in trigonometry. The cosine of sum of angles $a$ and $b$ is equal to the subtraction of the product of sines of both angles $a$ and $b$ from the product of cosines of angles $a$ and $b$.

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

This mathematical equation is called the cosine angle sum trigonometric identity in mathematics.

#### Usage

The cosine angle sum identity is used in two different cases in trigonometric mathematics.

##### Expansion

The cosine of sum of two angles is expanded as the subtraction of the product of sines of angles from the product of cosines of angles.

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

##### Simplification

The subtraction of the product of sines of angles from the product of cosines of angles is simplified as the cosine of sum of two angles.

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

#### Forms

The angle sum identity in cosine function can be expressed in several forms but the following 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 sum trigonometric identity by a geometric method in trigonometry.

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