Question

The value of $2ta{n}^{-1}(cosecta{n}^{-1}x-tanco{t}^{-1}x))$ is

• A. $ta{n}^{-1}x$
• B. $tanx$
• C. $cotx$
• D. $cose{c}^{-1}x$

Answer 1

Answer: (A)

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