Question

Show that 2tan^-1X=tan^-1 2X/1-X^2
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Answer 1

Given,

$2 \tan{}^{ - 1} x= \tan { }^{ - 1} \frac{2x}{1 - {x}^{2} }$

Now,

Let,

$tan { }^{ - 1}x = \alpha$

Then,

$tan \: \alpha = x$

We know that,

$tan \: 2 \alpha = \frac{2 \: tan \: \alpha }{1 - {tan}^{2} \alpha }$

$2 \: \alpha = {tan}^{ - 1} ( \frac{2 \: tan \: \alpha }{1 - {tan}^{2} \alpha } )$

Now,

We will substitute the value of tan alpha and alpha.

$2 {tan}^{ - 1} x = {tan}^{ - 1} ( \frac{2x}{1 - {x}^{2} } )$

Hence proved