One minus Cosine double angle identity

Formula

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

A trigonometric identity that expresses the subtraction of cosine of double angle from one as the two times square of sine of angle is called the one minus cosine double angle identity.

Introduction

When the theta ($\theta$) is used to denote an angle of a right triangle, the subtraction of cosine of double angle from one is written in the following mathematical form.

$1-\cos{2\theta}$

The subtraction of cosine of double angle from one is mathematically equal to the two times the sine squared of angle. It can be written in mathematical form as follows.

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

Usage

The one minus cosine of double angle identity is used as a formula in two cases in trigonometry.

Simplified form

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

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

Expansion

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

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

Other forms

The angle in the one minus cos double angle trigonometric identity can be denoted by any symbol. Hence, it also is popularly written in two distinct forms.

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

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

In this way, the one minus cosine of double angle formula can be expressed in terms of any symbol.

Proof

Learn how to prove the one minus cosine of double angle formula in trigonometric mathematics.