# Cos angle sum identity

### Expansion form

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

### Simplified form

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

### Introduction

The angle sum identity for the cosine function is generally called as cos of sum of two angles or cos of compound angle identity. It’s used as a trigonometric formula in two cases.

1. To expand cos of sum of two angles as the subtraction of product of cosines of angles from product of sines of angles.
2. To simplify the subtraction of products of cosines of angles from products of sines of angles as cos of compound angle function.

#### Formula

The angle sum cos formula is written in several ways in mathematics. For example, $\cos{(A+B)}$, $\cos{(x+y)}$, $\cos{(\alpha+\beta)}$, and so on.

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

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

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

#### Proof

Now, let’s learn how to derive the angle sum cos identity in mathematical form in trigonometry by geometrical method.

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