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.