Solve `cos^(-1) (cos x) gt sin^(-1) (sin x), x in [0, 2pi]`

Explanation

To solve the inequality \( \cos^{-1}(\cos x) > \sin^{-1}(\sin x) \) for \( x \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Understand the functions involved
The functions \( \cos^{-1}(\cos x) \) and \( \sin^{-1}(\sin x) \) have specific behaviors based on the values of \( x \).

### Step 2: Determine the ranges of the functions
- The function \( \cos^{-1}(\cos x) \) returns \( x \) if \( x \) is in the range \( [0, \pi] \) and returns \( 2\pi - x \) if \( x \) is in the range \( (\pi, 2\pi] \).
- The function \( \sin^{-1}(\sin x) \) returns \( x \) if \( x \) is in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) and returns \( \pi - x \) if \( x \) is in the range \( (\frac{\pi}{2}, \frac{3\pi}{2}] \) and \( 2\pi - x \) if \( x \) is in the range \( (\frac{3\pi}{2}, \frac{5\pi}{2}] \).
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