Evaluate \( \cos^{-1}(\cos -\frac{\pi}{4}) \)
Step-by-Step Solution
We need to evaluate:
\[ \cos^{-1}\left(\cos -\frac{\pi}{4}\right) \]
Step 1: Use identity
\[ \cos(-x) = \cos x \]
\[ \cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) \]
Step 2: Apply inverse cosine
\[ \cos^{-1}\left(\cos \frac{\pi}{4}\right) \]
The principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Since \( \frac{\pi}{4} \) lies in this interval, we get:
\[ \cos^{-1}\left(\cos -\frac{\pi}{4}\right) = \frac{\pi}{4} \]
Final Answer
\[ \boxed{\frac{\pi}{4}} \]