# Prove? sin2x=2tanx/1+tan^2x

Mar 21, 2018

As proved below.

#### Explanation:

$\text{To prove } \sin 2 x = \frac{2 \tan x}{1 + {\tan}^{2} x}$

Let us start from R H S and prove it equal to L H S.

$R H S = \frac{2 \tan x}{1 + {\tan}^{2} x}$

$\implies \frac{2 \tan x}{\sec} ^ 2 x , \text{ as " 1 + tan^2x = sec^2x, "identity}$

$\implies \frac{2 \left(\sin \frac{x}{\cancel{\cos}} x\right)}{\frac{1}{\cancel{{\cos}^{2} x}} ^ \textcolor{red}{\cos x}}$

$\implies 2 \sin x \cos x = \sin 2 x$ as per $\sin 2 x = 2 \sin x \cos x \text{ identity}$

$\implies L H S$