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. $cosec^{-1}\, x$

Answer 1

Answer: (A)

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