Statement:1: If `sum _(r=1)^(n) sin(x_(r)) = n`, then `sum_(n=1)^(n) cot (x_(r)) =n`. And Statement-2: The number of solutions of the equation `cosx =x` is 1.

Options

A. (a)Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-11
B. (b)Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-11
C. (c)Statement-1 is True, Statement-2 is false
D. (d)Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements provided and determine their validity.

### Step 1: Analyze Statement 1
The first statement is:
If \( \sum_{r=1}^{n} \sin(x_r) = n \), then \( \sum_{r=1}^{n} \cot(x_r) = n \).

1. From the equation \( \sum_{r=1}^{n} \sin(x_r) = n \), we can infer that each \( \sin(x_r) \) must be equal to 1, since the maximum value of \( \sin(x) \) is 1. This implies that:
\[ \sin(x_r) = 1 \quad \text{for all } r \]
Therefore, \( x_r = \frac{\pi}{2} + 2k\pi \) for some integer \( k \).

2. If \( \sin(x_r) = 1 \), then \( \cot(x_r) = \frac{\cos(x_r)}{\sin(x_r)} = \frac{0}{1} = 0 \).

3. Thus, we have:
\[ \sum_{r=1}^{n} \cot(x_r) = \sum_{r=1}^{n} 0 = 0 \]
This means that \( \sum_{r=1}^{n} \cot(x_r) \neq n \) unless \( n = 0 \).

**Conclusion for Statement 1**: The statement is false.

### Step 2: Analyze Statement 2
The second statement is:
The number of solutions of the equation \( \cos(x) = x \) is 1.

1. To analyze this, we can consider the functions \( y = \cos(x) \) and \( y = x \) graphically. The function \( \cos(x) \) oscillates between -1 and 1, while the line \( y = x \) is a straight line.

2. The function \( \cos(x) \) is continuous and decreases from 1 to -1 as \( x \) increases from 0 to \( \pi \).

3. The intersection points of \( y = \cos(x) \) and \( y = x \) can be found by observing that:
- At \( x = 0 \), \( \cos(0) = 1 \) and \( 0 < 1 \).
- At \( x = 1 \), \( \cos(1) \) is approximately 0.54, and \( 1 > 0.54 \).
- At \( x = \pi/2 \), \( \cos(\pi/2) = 0 \) and \( \pi/2 \approx 1.57 > 0 \).

4. Since \( \cos(x) \) is continuous and decreases from 1 to -1, and the line \( y = x \) is increasing, there will be exactly one intersection point in the interval \( [0, 1] \).

**Conclusion for Statement 2**: The statement is true.

### Final Conclusion
- Statement 1 is false.
- Statement 2 is true.

Thus, the correct option is that Statement 1 is false and Statement 2 is true.