Problem
Evaluate: \( \csc^{-1}(\csc \frac{\pi}{4}) \)
Solution
We know that:
\[ \csc \frac{\pi}{4} = \frac{1}{\sin \frac{\pi}{4}} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
So the expression becomes:
\[ \csc^{-1}(\sqrt{2}) \]
Now, recall the principal value range of \( \csc^{-1} x \):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
We need an angle whose cosecant is \( \sqrt{2} \).
Since:
\[ \csc \frac{\pi}{4} = \sqrt{2} \]
and \( \frac{\pi}{4} \) lies in the principal value range, we get:
\[ \csc^{-1}(\sqrt{2}) = \frac{\pi}{4} \]
Final Answer
\[ \boxed{\frac{\pi}{4}} \]