Evaluate \( \sec^{-1}(\sec \frac{9\pi}{5}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sec^{-1}\left(\sec \frac{9\pi}{5}\right) \]
Step 1: Reduce the angle
\[ \frac{9\pi}{5} = 2\pi – \frac{\pi}{5} \]
\[ \sec\left(\frac{9\pi}{5}\right) = \sec\left(2\pi – \frac{\pi}{5}\right) = \sec\left(\frac{\pi}{5}\right) \]
Step 2: Convert to cosine
\[ \sec \frac{\pi}{5} = \frac{1}{\cos \frac{\pi}{5}} \]
So we find angle with same sec value.
Step 3: Apply principal value range
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Step 4: Choose correct angle
\[ \sec \theta = \sec \frac{\pi}{5} \Rightarrow \theta = \frac{\pi}{5} \]
Since \( \frac{\pi}{5} \in [0, \pi] \), it is valid.
Final Answer
\[ \boxed{\frac{\pi}{5}} \]