If ^ -1 ( )=2 , then = - Vidyadip

MCQ

If $\tan^{-1}(\cot\theta)=2\theta,$ then $\theta=$

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

✓

Answer

Correct option: C.

$\pm\frac{\pi}{6}$

We have, $\tan^{-1}(\cot\theta)=2\theta$
$\Rightarrow\tan2\ \theta=\cot\theta$
$\Rightarrow\frac{2\tan\theta}{1-\tan^2\theta}=\frac{1}{\tan\theta}$
$\Rightarrow2\tan^2\theta=1-\tan^2\theta$
$\Rightarrow3\tan^2\theta=1$
$\Rightarrow\tan^2\theta=\frac{1}{3}$
$\Rightarrow\tan\theta=\pm\frac{1}{\sqrt3}$

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