STATEMENT -1 : `tan^(-1)x=sin^(-1)yrArry in (-1,1)`
and
STATEMENT -2 : `-pi/2 lt tan^(-1)x lt pi/2`

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 explanation 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 given statements, we need to analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** \( \tan^{-1} x = \sin^{-1} y \) implies that \( y \) belongs to the interval (-1, 1).

**Explanation:**
- The function \( \sin^{-1} y \) is defined for \( y \) in the interval [-1, 1]. However, the output of \( \sin^{-1} y \) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) when \( y \) takes values in (-1, 1).
- Since \( \tan^{-1} x \) also has a range of \(-\frac{\pi}{2} < \tan^{-1} x < \frac{\pi}{2}\), for the equality \( \tan^{-1} x = \sin^{-1} y \) to hold, \( y \) must be in the open interval (-1, 1) (not including -1 and 1).

### Step 2: Analyze Statement 2
**Statement 2:** \( -\frac{\pi}{2} < \tan^{-1} x < \frac{\pi}{2} \).

**Explanation:**
- This statement is true as it is the definition of the range of the \( \tan^{-1} \) function. The function \( \tan^{-1} x \) is defined for all real numbers \( x \) and its output always lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

### Step 3: Determine the Truth of Each Statement
- **Statement 1:** True, because \( \tan^{-1} x = \sin^{-1} y \) implies \( y \) must be in (-1, 1).
- **Statement 2:** True, as it correctly describes the range of the \( \tan^{-1} \) function.

### Step 4: Determine if Statement 2 Explains Statement 1
- Statement 2 provides the necessary information about the range of \( \tan^{-1} x \) which is crucial for understanding why \( y \) must be in (-1, 1). Thus, Statement 2 is indeed a correct explanation for Statement 1.

### Conclusion
Both statements are true, and Statement 2 is a correct explanation for Statement 1. Therefore, the correct option is:

**Option A:** Statement 1 is true, Statement 2 is true, and Statement 2 is the correct explanation for Statement 1.

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