Evaluate \( \tan^{-1}(\tan 2) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}(\tan 2) \]
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 \( 2 > \frac{\pi}{2} \), it lies outside the principal range.
Step 3: Use identity
\[ \tan(x – \pi) = \tan x \]
\[ 2 – \pi \]
Now check:
\[ 2 – \pi \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Step 4: Apply inverse tangent
\[ \tan^{-1}(\tan 2) = 2 – \pi \]
Final Answer
\[ \boxed{2 – \pi} \]