if the

Question:

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

(a) $\pm \frac{\pi}{3}$

(b) $\pm \frac{\pi}{4}$

(c) $\pm \frac{\pi}{6}$

(d) none of these

Solution:

(c) $\pm \frac{\pi}{6}$

We have,

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

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

$\Rightarrow \frac{2 \tan \theta}{1-\tan ^{2} \theta}=\frac{1}{\tan \theta}$

$\Rightarrow 2 \tan ^{2} \theta=1-\tan ^{2} \theta$

$\Rightarrow 3 \tan ^{2} \theta=1$

$\Rightarrow \tan ^{2} \theta=\frac{1}{3}$

$\Rightarrow \tan \theta=\pm \frac{1}{\sqrt{3}}$

$\therefore \theta=\pm \frac{\pi}{6}$

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