Evaluate \( \tan^{-1}(\tan \frac{9\pi}{4}) \)
Step-by-Step Solution
We need to evaluate:
\[ \tan^{-1}\left(\tan \frac{9\pi}{4}\right) \]
Step 1: Use periodicity
\[ \tan(x + \pi) = \tan x \]
\[ \frac{9\pi}{4} = 2\pi + \frac{\pi}{4} \Rightarrow \tan\left(\frac{9\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) \]
Step 2: Apply inverse tangent
\[ \tan^{-1}\left(\tan \frac{\pi}{4}\right) \]
The principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Since \( \frac{\pi}{4} \) lies in this range, we get:
\[ \tan^{-1}(\tan \frac{9\pi}{4}) = \frac{\pi}{4} \]
Final Answer
\[ \boxed{\frac{\pi}{4}} \]