Math Doubts

Cosine double angle identity

Formula

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

A trigonometric identity that expresses the expansion of cosine of double angle in cosine and sine of angle is called the cosine of double angle identity.

Introduction

When the angle of a right triangle is denoted by a symbol theta, the cosine and sine of angle are written as $\cos{\theta}$ and $\sin{\theta}$ respectively. In the same way, the cosine of double angle is written in mathematical form as $\cos{2\theta}$ in trigonometry.

The cosine of double angle can be written in terms of sine and cosine of angle in subtraction form as follows.

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

It is called the cos double angle identity and used as a formula in trigonometric mathematics. It can written mathematically in two forms.

Cosine form

The cosine of double angle rule is purely expressed in cosine form as follows.

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

Sine form

The cosine of double angle rule is also purely expressed in sine form as follows.

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

Usage

The cosine of double angle identity is mostly used in two different cases in the trigonometry.

Expansion

It is used to expand the sine of double angle functions in sine and cosine functions.

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

Simplified form

It is used to simplify the difference of sine and cosine functions as cosine of double angle function.

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

Other forms

The angle in cosine of double angle formula can be represented by any symbol. So, the cosine of double angle identity can be expressed in terms of any variable. It is usually written in three other popular forms.

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

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

$(3). \,\,\,\,\,\,$ $\cos{2\alpha}$ $\,=\,$ $\cos^2{\alpha}-\sin^2{\alpha}$

Proof

Learn how to derive the rule for the cosine double angle in trigonometry by geometric method.

Another form

The cosine of double angle formula can also be expanded in terms of tan of the angle in mathematics.

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

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