`f(x)=sin^(-1)(sin x), g(x)=cos^(-1)(cos x)`, then :

Options

A. `f(x)=g(x)" if " x in (0, (pi)/(4))`
B. `f(x) lt g(x)" if " x in ((pi)/(2), (3pi)/(4))`
C. `f(x) lt g(x)" if " (pi, (5pi)/(4))`
D. `f(x) gt g(x)" if " x in (pi, (5pi)/(4))`

Answer

To solve the problem, we need to analyze the functions \( f(x) = \sin^{-1}(\sin x) \) and \( g(x) = \cos^{-1}(\cos x) \) over specific intervals. We will determine the values of these functions within the intervals \( [0, \frac{\pi}{2}] \), \( [\frac{\pi}{2}, \pi] \), and \( [\pi, \frac{3\pi}{2}] \).

### Step 1: Determine the range of the inverse trigonometric functions
The range of \( \sin^{-1}(x) \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) and the range of \( \cos^{-1}(x) \) is \( [0, \pi] \).

### Step 2: Analyze the function \( f(x) = \sin^{-1}(\sin x) \)
1. **Interval \( [0, \frac{\pi}{2}] \)**:
- In this interval, \( \sin x \) is non-negative and \( \sin^{-1}(\sin x) = x \).
- Thus, \( f(x) = x \).

2. **Interval \( [\frac{\pi}{2}, \pi] \)**:
- Here, \( \sin x \) is still non-negative, but \( \sin^{-1}(\sin x) \) will give us \( \pi - x \) since \( \sin x \) is symmetric about \( \frac{\pi}{2} \).
- Thus, \( f(x) = \pi - x \).

3. **Interval \( [\pi, \frac{3\pi}{2}] \)**:
- In this interval, \( \sin x \) is negative, and thus \( \sin^{-1}(\sin x) = \pi - x \) (as \( \sin x \) is negative).
- Thus, \( f(x) = \pi - x \).

### Step 3: Analyze the function \( g(x) = \cos^{-1}(\cos x) \)
1. **Interval \( [0, \frac{\pi}{2}] \)**:
- In this interval, \( \cos x \) is non-negative, so \( g(x) = x \).

2. **Interval \( [\frac{\pi}{2}, \pi] \)**:
- Here, \( \cos x \) is non-positive, and \( g(x) = \pi - x \).

3. **Interval \( [\pi, \frac{3\pi}{2}] \)**:
- In this interval, \( \cos x \) is negative, and thus \( g(x) = 2\pi - x \).

### Step 4: Summarize the results in a table
| Interval | \( f(x) \) | \( g(x) \) |
|-----------------------|--------------------|---------------------|
| \( [0, \frac{\pi}{2}] \) | \( x \) | \( x \) |
| \( [\frac{\pi}{2}, \pi] \) | \( \pi - x \) | \( \pi - x \) |
| \( [\pi, \frac{3\pi}{2}] \) | \( \pi - x \) | \( 2\pi - x \) |

### Step 5: Analyze the options
1. **Option A**: \( f(x) = g(x) \) for \( x \in [0, \frac{\pi}{2}] \) - **True**.
2. **Option B**: \( f(x) < g(x) \) for \( x \in [\frac{\pi}{2}, \pi] \) - **True** since \( \pi - x < x \).
3. **Option C**: \( f(x) < g(x) \) for \( x \in [\pi, \frac{5\pi}{4}] \) - **True** since \( \pi - x < 2\pi - x \).
4. **Option D**: This option is incorrect since options A, B, and C are correct.

### Final Conclusion
The correct options are A, B, and C.