Problem
Evaluate: \( \csc^{-1}(\csc \frac{6\pi}{5}) \)
Solution
First, note that:
\[ \frac{6\pi}{5} = \pi + \frac{\pi}{5} \]
Using identity:
\[ \sin(\pi + \theta) = -\sin \theta \]
So,
\[ \sin \frac{6\pi}{5} = -\sin \frac{\pi}{5} \]
Thus,
\[ \csc \frac{6\pi}{5} = \frac{1}{\sin \frac{6\pi}{5}} = -\frac{1}{\sin \frac{\pi}{5}} = -\csc \frac{\pi}{5} \]
Now the expression becomes:
\[ \csc^{-1}(-\csc \frac{\pi}{5}) \]
Recall the principal value range of \( \csc^{-1} x \):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Since the value is negative, the angle must lie in:
\[ \left[-\frac{\pi}{2}, 0\right) \]
Also,
\[ \csc\left(-\frac{\pi}{5}\right) = -\csc \frac{\pi}{5} \]
And \( -\frac{\pi}{5} \) lies within the principal value range.
Final Answer
\[ \boxed{-\frac{\pi}{5}} \]