Problem
Evaluate: \( \cot^{-1}(\cot(-\frac{8\pi}{3})) \)
Solution
First, reduce the angle using periodicity of cotangent (period \( \pi \)):
\[ -\frac{8\pi}{3} = -3\pi + \frac{\pi}{3} \]
So,
\[ \cot\left(-\frac{8\pi}{3}\right) = \cot\left(\frac{\pi}{3}\right) \]
Now,
\[ \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \]
Thus 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
\[ \boxed{\frac{\pi}{3}} \]