Let `f(x)=sin^(-1)x and g(x)=cos^(-1)x`, then which of the following statements are correct?

Options

A. `{:"f(x)" gt"g(x),"if"x in (1/sqrt2,1]:}`
B. `{:"f(x)" lt"g(x),"if"x in [-1,1/sqrt2):}`
C. `{:"f(x)" gt"g(x),"if"x in (-1/sqrt2,1/sqrt2):}`
D. `{:"f(x)" lt"g(x),"if"x in [-1/sqrt2,1/sqrt2]:}`

Answer

To solve the problem, we need to analyze the functions \( f(x) = \sin^{-1}(x) \) and \( g(x) = \cos^{-1}(x) \) and determine the validity of the given statements regarding their relationships over specified intervals.

### Step-by-Step Solution:

1. **Identify the Functions and Their Ranges**:
- The function \( f(x) = \sin^{-1}(x) \) has a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
- The function \( g(x) = \cos^{-1}(x) \) has a range of \( [0, \pi] \).

2. **Determine the Intervals of Interest**:
- We need to evaluate the relationships between \( f(x) \) and \( g(x) \) over the intervals:
- \( \left[\frac{1}{\sqrt{2}}, 1\right] \)
- \( \left[-1, \frac{1}{\sqrt{2}}\right] \)
- \( \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \)

3. **Analyze the Functions on Each Interval**:
- **Interval \( \left[\frac{1}{\sqrt{2}}, 1\right] \)**:
- In this interval, \( f(x) \) is increasing and \( g(x) \) is decreasing.
- Therefore, \( f(x) > g(x) \) is true.

- **Interval \( \left[-1, \frac{1}{\sqrt{2}}\right] \)**:
- In this interval, \( f(x) \) is still increasing while \( g(x) \) is decreasing.
- Hence, \( f(x) < g(x) \) is true.

- **Interval \( \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \)**:
- At \( x = -\frac{1}{\sqrt{2}} \), \( f(x) < g(x) \).
- At \( x = \frac{1}{\sqrt{2}} \), \( f(x) = g(x) \).
- Therefore, \( f(x) > g(x) \) is false, and \( f(x) < g(x) \) is true except at the point \( x = \frac{1}{\sqrt{2}} \).

4. **Conclusion**:
- The correct statements are:
- \( f(x) > g(x) \) for \( x \in \left[\frac{1}{\sqrt{2}}, 1\right] \) (True)
- \( f(x) < g(x) \) for \( x \in \left[-1, \frac{1}{\sqrt{2}}\right] \) (True)
- \( f(x) > g(x) \) for \( x \in \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \) (False)
- \( f(x) < g(x) \) for \( x \in \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \) (True, but not strictly less than at \( x = \frac{1}{\sqrt{2}} \))

### Final Answer:
The correct statements are:
1. \( f(x) > g(x) \) for \( x \in \left[\frac{1}{\sqrt{2}}, 1\right] \)
2. \( f(x) < g(x) \) for \( x \in \left[-1, \frac{1}{\sqrt{2}}\right] \)