Statement-1: `sinx = siny rArr x =y`.
and Statement-2: `sinx = siny` have infinitely many solutions for real values of `x` and y.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-1
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the question, we need to analyze both statements provided:

**Statement-1:** \( \sin x = \sin y \Rightarrow x = y \)

**Statement-2:** \( \sin x = \sin y \) has infinitely many solutions for real values of \( x \) and \( y \).

### Step 1: Analyze Statement-1

1. The equation \( \sin x = \sin y \) does not imply that \( x \) must equal \( y \).
2. The sine function is periodic, meaning it repeats its values. Specifically, \( \sin x = \sin y \) can occur for multiple values of \( x \) and \( y \) that are not equal.
3. The general solution for \( \sin x = \sin y \) is given by:
\[ x = n\pi + (-1)^n y \]
where \( n \) is any integer. This shows that there are infinitely many pairs of \( (x, y) \) that satisfy the equation without requiring \( x \) to equal \( y \).

### Conclusion for Statement-1:
- Since \( \sin x = \sin y \) does not imply \( x = y \), **Statement-1 is false**.

### Step 2: Analyze Statement-2

1. From the analysis above, we see that \( \sin x = \sin y \) has infinitely many solutions.
2. For any integer \( n \), we can find corresponding \( x \) and \( y \) values using the general solution:
\[ x = n\pi + (-1)^n y \]
3. This means for any chosen value of \( y \), we can find multiple \( x \) values, confirming that there are indeed infinitely many solutions.

### Conclusion for Statement-2:
- Since \( \sin x = \sin y \) does have infinitely many solutions, **Statement-2 is true**.

### Final Conclusion:
- Statement-1 is false, and Statement-2 is true.

### Answer:
The correct option is 4 (Statement-1 is false, Statement-2 is true).

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