Evaluate \( \tan^{-1}(\tan 1) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}(\tan 1) \]
Step 1: Principal value range
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 2: Check the angle
Since \( 1 \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), it already lies in the principal range.
Step 3: Apply inverse tangent
\[ \tan^{-1}(\tan 1) = 1 \]
Final Answer
\[ \boxed{1} \]