Sine double angle identity in Tan function

Formula

$\sin{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

A trigonometric identity that expresses the expansion of sine of double angle function in terms of tan function is called the sine of double angle identity in tangent function.

Introduction

When the symbol theta represents an angle of a right triangle, the sine and tangent functions are written as $\sin{\theta}$ and $\tan{\theta}$ respectively. In the same way, the sine of double angle function is written as $\sin{(2\theta)}$ mathematically.

The sine double angle function can be expressed in terms of tan functions in the following rational form.

$\implies$ $\sin{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

Usage

The sine of double angle formula in terms of tan function is used in two cases in trigonometric mathematics.

Expansion

It is used to expand the sine of double angle function in terms of tan of angle function.

$\implies$ $\sin{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

Simplified form

It is also used to simplify the following form rational expression as the sine of double angle function.

$\implies$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$ $\,=\,$ $\sin{2\theta}$

Other forms

The angle in the sin double angle formula can be denoted by any symbol. Therefore, the following three are popular forms of sine of double angle identity in terms of tangent.

$(1)\,\,\,\,\,\,$ $\sin{2x}$ $\,=\,$ $\dfrac{2\tan{x}}{1+\tan^2{x}}$

$(2)\,\,\,\,\,\,$ $\sin{2A}$ $\,=\,$ $\dfrac{2\tan{A}}{1+\tan^2{A}}$

$(3)\,\,\,\,\,\,$ $\sin{2\alpha}$ $\,=\,$ $\dfrac{2\tan{\alpha}}{1+\tan^2{\alpha}}$