Question:
Find the principal value of:
\[ \sin^{-1}\left(\tan \frac{5\pi}{4}\right) \]
Solution:
Step 1: Evaluate tangent value
\[ \tan \frac{5\pi}{4} = 1 \]
Step 2: Substitute
\[ \sin^{-1}(1) \]
Step 3: Use standard value
\[ \sin^{-1}(1) = \frac{\pi}{2} \]
Step 4: Check principal range
\[ \frac{\pi}{2} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
—Final Answer:
\[ \boxed{\frac{\pi}{2}} \]