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Question
If tan−1 (cot θ) = 2 θ, then θ =
Options
`+-pi/3`
`+-pi/4`
`+-pi/6`
none of these
Solution
(c) `+-pi/6`
\[\text{We have}, \]
\[ \tan^{- 1} \left( cot\theta \right) = 2\theta\]
\[ \Rightarrow \tan2\theta = cot\theta\]
\[ \Rightarrow \frac{2\tan\theta}{1 - \tan^2 \theta} = \frac{1}{\tan\theta}\]
\[ \Rightarrow 2 \tan^2 \theta = 1 - \tan^2 \theta\]
\[ \Rightarrow 3 \tan^2 \theta = 1\]
\[ \Rightarrow \tan^2 \theta = \frac{1}{3}\]
\[ \Rightarrow \tan\theta = \pm \frac{1}{\sqrt{3}}\]
\[ \therefore \theta = \pm \frac{\pi}{6}\]
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