Question
If
\[ \tan^{-1}(\cot\theta) = 2\theta \]
Find \( \theta \).
Solution
Use identity:
\[ \cot\theta = \tan\left(\frac{\pi}{2} – \theta\right) \]
So,
\[ \tan^{-1}(\cot\theta) = \tan^{-1}\left(\tan\left(\frac{\pi}{2} – \theta\right)\right) \]
Using principal value:
\[ \tan^{-1}(\tan x) = x \quad \text{if } x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Thus,
\[ \tan^{-1}(\cot\theta) = \frac{\pi}{2} – \theta \]
Given:
\[ \frac{\pi}{2} – \theta = 2\theta \]
\[ \frac{\pi}{2} = 3\theta \Rightarrow \theta = \frac{\pi}{6} \]
Final Answer:
\[ \boxed{\frac{\pi}{6}} \]
Key Concept
Convert cot into tan form and apply principal value condition carefully.