Which of the following statement(s) is/are correct?

Options

A. `cos(sin 1) gt sin( cos 1)`
B. `cos(sin 1.5) > sin( cos 1.5)`
C. `cos(sin (7pi)/18) gt sin(cos (7pi)/18)`
D. `cos(sin (5pi)/18) gt sin(cos (5pi)/18)`

Answer

To solve the problem, we need to analyze the functions \( f(x) = \cos(\sin x) \) and \( g(x) = \sin(\cos x) \) and determine which statements about them are correct.

### Step-by-Step Solution:

1. **Define the Functions**:
- Let \( f(x) = \cos(\sin x) \)
- Let \( g(x) = \sin(\cos x) \)

2. **Evaluate at \( x = 0 \)**:
- Calculate \( f(0) \):
\[ f(0) = \cos(\sin(0)) = \cos(0) = 1 \]
- Calculate \( g(0) \):
\[ g(0) = \sin(\cos(0)) = \sin(1) \]

3. **Compare \( f(0) \) and \( g(0) \)**:
- We know that \( \sin(1) < 1 \) because \( \sin(\frac{\pi}{2}) = 1 \) and \( 1 < \frac{\pi}{2} \approx 1.57 \).
- Thus, we have:
\[ f(0) = 1 > \sin(1) = g(0) \]

4. **Consider the General Case**:
- We need to check if \( f(x) > g(x) \) for all \( x < \frac{\pi}{2} \).
- Set the equation \( f(x) = g(x) \):
\[ \cos(\sin x) = \sin(\cos x) \]

5. **Analyze the Equation**:
- Rewrite \( \cos(\sin x) \) using the identity \( \cos(a) = \sin\left(\frac{\pi}{2} - a\right) \):
\[ \cos(\sin x) = \sin\left(\frac{\pi}{2} - \sin x\right) \]
- Thus, we have:
\[ \sin\left(\frac{\pi}{2} - \sin x\right) = \sin(\cos x) \]

6. **Check for Intersection Points**:
- The maximum value of \( \cos(\sin x) \) is 1, and the maximum value of \( \sin(\cos x) \) is also less than or equal to 1.
- The functions \( f(x) \) and \( g(x) \) can intersect at points, but we need to analyze their behavior in the interval \( x < \frac{\pi}{2} \).

7. **Conclusion**:
- Since \( f(0) > g(0) \) and both functions are continuous, we can conclude that \( f(x) > g(x) \) for all \( x \) in the interval \( [0, \frac{\pi}{2}) \).
- Therefore, all statements regarding \( f(x) \) and \( g(x) \) being compared are correct.

### Final Answer:
All the statements are correct.