Cosine double angle identity in square of Sine

Formula

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

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

Introduction

Let theta be an angle of a right triangle. The cosine of double angle and sine squared of angle are written as $\cos{2\theta}$ and $\sin^2{\theta}$ respectively.

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

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

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

Usage

In trigonometric mathematics, it is used as a formula possibly in two different cases.

Expansion

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

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

Simplified form

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

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

Other forms

The angle in cosine of double angle formula in terms of square of sine function can be denoted by any symbol. Hence, it is written in two other forms popularly.

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

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

Proof

Learn how to derive the mathematical rule for the cosine of double angle in sine squared function in trigonometry.