Consider the following statements S and R:
S: Both sin x and cos x are decreasing functions in the interval `(pi//2, pi)`
R: If a differentiable function decreases in an interval [a,b), then its derivative also decreases in [a,b). Which of the following is true ?

Options

A. both S and R are wrong.
B. both S and R are correct, but R is not the correct explanation for S.
C. S is correct and R is the correct explanation for S.
D. S is correct and R is wrong.

Answer

To solve the problem, we need to analyze the two statements S and R regarding the functions sin x and cos x.

### Step 1: Analyze Statement S
**Statement S:** Both sin x and cos x are decreasing functions in the interval \( \left( \frac{\pi}{2}, \pi \right) \).

1. **Evaluate sin x:**
- The function sin x is defined for all real numbers and is known to be increasing in the interval \( \left( 0, \frac{\pi}{2} \right) \) and decreasing in the interval \( \left( \frac{\pi}{2}, \pi \right) \).
- At \( x = \frac{\pi}{2} \), sin x reaches its maximum value of 1, and as x approaches π, sin x decreases to 0.

2. **Evaluate cos x:**
- The function cos x is defined for all real numbers and is known to be decreasing in the interval \( \left( 0, \pi \right) \).
- At \( x = \frac{\pi}{2} \), cos x reaches its minimum value of 0 and continues to decrease to -1 at \( x = \pi \).

3. **Conclusion for Statement S:**
- Both sin x and cos x are indeed decreasing in the interval \( \left( \frac{\pi}{2}, \pi \right) \).
- Therefore, Statement S is **True**.

### Step 2: Analyze Statement R
**Statement R:** If a differentiable function decreases in an interval \([a, b)\), then its derivative also decreases in \([a, b)\).

1. **Consider the function sin x:**
- We know that sin x is decreasing in the interval \( \left( \frac{\pi}{2}, \pi \right) \).
- The derivative of sin x is \( f'(x) = \cos x \).
- In the interval \( \left( \frac{\pi}{2}, \pi \right) \), cos x is decreasing from 0 to -1.

2. **Consider the interval \( \left( \pi, \frac{3\pi}{2} \right) \):**
- In this interval, sin x is also decreasing.
- However, the derivative \( f'(x) = \cos x \) is increasing from -1 to 0.

3. **Conclusion for Statement R:**
- While it is true that the derivative of sin x is decreasing in the interval \( \left( \frac{\pi}{2}, \pi \right) \), it is not necessarily true for all intervals where the function is decreasing.
- Therefore, Statement R is **False**.

### Final Conclusion
- Statement S is True.
- Statement R is False.

Thus, the correct answer is that Statement S is correct and Statement R is incorrect.