Question

The value of $2{\tan }^{-1}(cosec{\tan }^{-1}x-\tan {\cot }^{-1}x)$ is

• A. ${\tan }^{-1}x$
• B. $\tan x$
• C. $\cot x$
• D. $cose{c}^{-1}x$

Answer 1

Answer: (A)

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

$2{\tan }^{-1}(cosec{\tan }^{-1}x-\tan {\cot }^{-1}x)$
$=2{\tan }^{-1}[cosec\left\{cose{c}^{-1}\frac{\sqrt{1+x^2}}{x}\right\}-\tan \left\{{\tan }^{-1}\frac{1}{x}\right\}]$
$=2{\tan }^{-1}[\frac{\sqrt{1+x^2}}{x}-\frac{1}{x}]$
$=2{\tan }^{-1}\left\{\frac{\sqrt{1+x^2}-1}{x}\right\}$
$=2{\tan }^{-1}\left\{\frac{\sec \theta -1}{\tan \theta }\right\}(putx=\tan \theta )$
$=2{\tan }^{-1}\left\{\frac{1-\cos \theta }{\sin \theta }\right\}$
$=2{\tan }^{-1}\left\{\frac{2{\sin }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cdot \cos \frac{\theta }{2}}\right\}$
$=2{\tan }^{-1}\tan \frac{\theta }{2}$
$=2\cdot \frac{\theta }{2}=\theta ={\tan }^{-1}x$

Hence, option A is correct.