Math Doubts

Cosine double angle identity in square of Cosine

Formula

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

A mathematical identity that expresses the expansion of cosine of double angle in terms of cosine squared of angle is called the cosine of double angle identity in square of cosine.

Introduction

When the theta represents an angle of a right triangle, the cosine of double angle and cosine squared of angle are written as $\cos{2\theta}$ and $\cos^2{\theta}$ respectively.

The cosine of double angle is equal to the subtraction of one from two times the square of cosine.

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

It is called the cosine of double angle identity in terms of cosine squared form.

Usage

In trigonometry, it can be used as a formula in two distinct cases.

Expansion

It is used to expand the cosine of double angle functions as the subtraction of one from two times the cosine squared of angle.

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

Simplified form

It is also used to simplify the subtraction of one from two times the cosine squared of angle as the cosine of double angle function.

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

Other forms

The angle in cos of double angle formula can be represented by any symbol. Therefore, it is popularly written in two other forms.

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

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

Proof

Learn how to derive the rule for the cosine of double angle in terms of square of cosine function in trigonometry.

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