Problem
Evaluate: \( \cot^{-1}(\cot \frac{\pi}{3}) \)
Solution
We know that:
\[ \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \]
So the expression becomes:
\[ \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) \]
Recall the principal value range of \( \cot^{-1} x \):
\[ (0, \pi) \]
We need an angle in this range whose cotangent is \( \frac{1}{\sqrt{3}} \).
We know that:
\[ \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \]
And \( \frac{\pi}{3} \) lies in the principal value range.
Final Answer
\[ \boxed{\frac{\pi}{3}} \]