Sin double angle formula in terms of tan of angle

Sin double angle formula is used to expand double angle sine functions such as $\sin{2x}$, $\sin{2A}$, $\sin{2\alpha}$, $\sin{2\theta}$ and etc. in terms of sine and cosine of angle. The same sine double angle rule can also be expressed in terms of tan function and it is required in some cases. So, it is very important to learn how to express sin double angle identity in terms of tan function.

Formula

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

Other form

Sine double angle formula in terms of tan of angle is also usually written in the following three forms.

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

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

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

Proof

Sin double angle formula in terms of tan function can be derived in trigonometric mathematics by combining some trigonometric formulas.