Question

Solve :

$\frac{\left(2\tan x\right)}{(1+{\tan }^{2}x)}$

Answer 1

$\Rightarrow$ $1+{\tan }^{2}x=se{c}^{2}x$

$\Rightarrow$ $\tan x=\sin x/\cos x$

$$\begin{gathered} =\frac{{\displaystyle \frac{2\sin x}{\cos x}}}{se{c}^{2}x} \\ =\frac{\frac{2\sin x}{\cos x}}{{\displaystyle \frac{1}{{\cos }^{2}x}}} \end{gathered}$$

$$\begin{gathered} =\frac{2\sin x}{\cos x}\times {\cos }^{2}x \\ =2\sin x\cos x \end{gathered}$$

$=\sin (2x)$

Hence, $\sin (2x)$ is the correct answer.