Question

prove that 2tan-1x =tan-1(2x/1-x^2)​

Answer 1

We know that,

$\tan(2\theta)=\dfrac {2\tan\theta}{1-\tan^2\theta}$

Taking inverse on both sides, we get,

$2\theta=\tan^{-1}\left (\dfrac {2\tan\theta}{1-\tan^2\theta}\right)\quad\longrightarrow\quad(1)$

Let $x=\tan\theta.$

Then $\tan^{-1}x=\theta$

Hence (1) becomes,

$2\tan^{-1}x=\tan^{-1}\left (\dfrac {2x}{1-x^2}\right)$

Hence Proved!