Assertion (A) : `tan^(-1)x` is `((-pi)/(2),(pi)/(2))`
Reason (R ) : Domain of `tan^(-1)x` is R.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the assertion and reason provided in the question, we will analyze both statements step by step.

### Step 1: Understand the Assertion (A)
The assertion states that the range of the function \( \tan^{-1}(x) \) (the inverse tangent function) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

- **Explanation**: The function \( \tan^{-1}(x) \) takes any real number \( x \) as input and outputs an angle \( y \) such that \( \tan(y) = x \). The angles \( y \) produced by \( \tan^{-1}(x) \) are confined to the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which means that the assertion is true.

### Step 2: Understand the Reason (R)
The reason states that the domain of \( \tan^{-1}(x) \) is \( \mathbb{R} \) (the set of all real numbers).

- **Explanation**: The function \( \tan^{-1}(x) \) can accept any real number as input. Therefore, the domain of \( \tan^{-1}(x) \) is indeed all real numbers, which makes this statement true as well.

### Step 3: Analyze the Relationship Between Assertion and Reason
Now we need to determine if the reason provided is a correct explanation for the assertion.

- The assertion is about the range of \( \tan^{-1}(x) \), while the reason discusses its domain. Although both statements are true, the reason does not explain why the range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Conclusion
Both the assertion and the reason are true, but the reason does not correctly explain the assertion. Therefore, the correct conclusion is that both statements are true, but the reason is not the correct explanation for the assertion.

### Final Answer
- Assertion (A) is true.
- Reason (R) is true.
- However, R is not the correct explanation for A.