Consider the following statements :
1. The function f(x)=sinx decreases on the interval `(0,pi//2)`.
2. The function f(x)=cosx increases on the interval `(0,pi//2)`.
Which of the above statements is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the two statements about the functions \( f(x) = \sin x \) and \( f(x) = \cos x \) over the interval \( (0, \frac{\pi}{2}) \).

### Step-by-Step Solution:

1. **Analyze the first statement: \( f(x) = \sin x \) decreases on the interval \( (0, \frac{\pi}{2}) \)**.
- The sine function is known to increase in the interval \( (0, \frac{\pi}{2}) \).
- To confirm this, we can evaluate the sine function at specific points:
- \( \sin(0) = 0 \)
- \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)
- \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \approx 0.707 \)
- \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \approx 0.866 \)
- \( \sin\left(\frac{\pi}{2}\right) = 1 \)
- As we can see, the values of \( \sin x \) are increasing from \( 0 \) to \( 1 \) as \( x \) goes from \( 0 \) to \( \frac{\pi}{2} \).
- Therefore, the first statement is **incorrect**.

2. **Analyze the second statement: \( f(x) = \cos x \) increases on the interval \( (0, \frac{\pi}{2}) \)**.
- The cosine function is known to decrease in the interval \( (0, \frac{\pi}{2}) \).
- To confirm this, we can evaluate the cosine function at specific points:
- \( \cos(0) = 1 \)
- \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.866 \)
- \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \approx 0.707 \)
- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- As we can see, the values of \( \cos x \) are decreasing from \( 1 \) to \( 0 \) as \( x \) goes from \( 0 \) to \( \frac{\pi}{2} \).
- Therefore, the second statement is also **incorrect**.

### Conclusion:
Both statements are incorrect.