Question

If ${\tan }^{-1}\left(\cot \theta \right)=2\theta$, then find the value of $\theta$

• A. $\pm \frac{\pi }{3}$
• B. $\pm \frac{\pi }{4}$
• C. $\pm \frac{\pi }{6}$
• D. None of these

Answer 1

The correct option is C $\pm \frac{\pi }{6}$

given, ${\tan }^{-1}\left(\cot \theta \right)=2\theta$

$\Rightarrow \tan 2\theta =\cot \theta$

$\Rightarrow \tan 2\theta =\tan (\frac{\pi }{2}-\theta )$ OR $\tan 2\theta =\tan (-\frac{\pi }{2}-\theta )$

$\Rightarrow 2\theta =\frac{\pi }{2}-\theta$ OR $2\theta =-\frac{\pi }{2}-\theta$

$\Rightarrow 3\theta =\pm \frac{\pi }{2}$

$\Rightarrow \theta =\pm \frac{\pi }{6}$
$\text{OR}$

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

$\Rightarrow \tan 2\theta =\cot \theta$
$\Rightarrow \frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{1}{\tan \theta }$
$\Rightarrow 2{\tan }^2\theta =1-{\tan }^2\theta$
$\Rightarrow {\tan }^2\theta =\frac{1}{3}$
$\Rightarrow \tan \theta =\pm \frac{1}{\sqrt{3}}$
$\therefore \theta =\pm \frac{\pi }{6}$