Write the function in the simplest form: ^ -1 1+x^ 2 -1 x , x 0 - Vidyadip

MCQ

Write the function in the simplest form: $\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0$

A
$\frac{1}{2} \sin ^{-1} x$
B
$ \tan ^{-1} x$
C
$\frac{1}{2} \cot ^{-1} x$
✓
$\frac{1}{2} \tan ^{-1} x$

✓

Answer

Correct option: D.

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

d
$\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}$

Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$

$\therefore \tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}=\tan ^{-1} \frac{\sqrt{1+\tan ^{2} \theta}-1}{\tan \theta}$

$=\tan ^{-1}\left(\frac{\sec \theta-1}{\tan \theta}\right)=\tan ^{-1}\left(\frac{1-\cos \theta}{\sin \theta}\right)$

$\tan ^{-1}\left(\frac{2 \sin ^{2}\left(\frac{\theta}{2}\right)}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}\right)$

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

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