One plus Cosine double angle identity

Formula

$1+\cos{(2\theta)} \,=\, 2\cos^2{\theta}$

A trigonometric identity that expresses the addition of one and cosine of double angle as the two times square of cosine of angle is called the one plus cosine double angle identity.

Introduction

If the theta ($\theta$) is used to represent an angle of a right triangle, the sum of one and cosine of double angle is mathematically written as follows.

$1+\cos{2\theta}$

The sum of one and cosine of double angle is mathematically equal to the two times the cosine squared of angle. It can be expressed in mathematical form as follows.

$\implies$ $1+\cos{(2\theta)}$ $\,=\,$ $2\cos^2{\theta}$

Usage

The one plus cosine of double angle identity is mostly used as a formula in two different cases in the trigonometry.

Simplified form

It is used to simplify the one plus cos of double angle as two times the square of cosine of angle.

$\implies$ $1+\cos{(2\theta)} \,=\, 2\cos^2{\theta}$

Expansion

It is used to expand the two times cos squared of angle as the one plus cosine of double angle.

$\implies$ $2\cos^2{\theta} \,=\, 1+\cos{(2\theta)}$

Other forms

The angle in the one plus cos double angle trigonometric identity can be represented by any symbol but it is popularly written in two different forms

$(1). \,\,\,$ $1+\cos{(2x)} \,=\, 2\cos^2{x}$

$(2). \,\,\,$ $1+\cos{(2A)} \,=\, 2\cos^2{A}$

Thus, the one plus cosine of double angle rule can be written in terms of any symbol.

Proof

Learn how to derive the one plus cosine of double angle trigonometric identity in trigonometry.