Question
If
\[ \tan^{-1}(\sqrt{3}) + \cot^{-1}x = \frac{\pi}{2} \]
Find \( x \).
Solution
We know:
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]
So,
\[ \frac{\pi}{3} + \cot^{-1}x = \frac{\pi}{2} \]
\[ \cot^{-1}x = \frac{\pi}{2} – \frac{\pi}{3} = \frac{\pi}{6} \]
Now,
\[ \cot^{-1}x = \frac{\pi}{6} \Rightarrow x = \cot\left(\frac{\pi}{6}\right) \]
\[ = \sqrt{3} \]
Final Answer:
\[ \boxed{\sqrt{3}} \]
Key Concept
Convert inverse trigonometric expressions into standard angles for easy solving.