Problem
Evaluate: \( \csc^{-1}(\csc \frac{3\pi}{4}) \)
Solution
We know that:
\[ \sin \frac{3\pi}{4} = \frac{1}{\sqrt{2}} \]
So,
\[ \csc \frac{3\pi}{4} = \frac{1}{\sin \frac{3\pi}{4}} = \sqrt{2} \]
Thus the expression becomes:
\[ \csc^{-1}(\sqrt{2}) \]
Now recall the principal value range of \( \csc^{-1} x \):
\[ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \]
Although \( \csc \frac{3\pi}{4} = \sqrt{2} \), the angle \( \frac{3\pi}{4} \) is not in the principal value range.
The angle in the principal range whose cosecant is \( \sqrt{2} \) is:
\[ \frac{\pi}{4} \]
Final Answer
\[ \boxed{\frac{\pi}{4}} \]