If `(1)/(2) lt |x| lt 1`, then which of the following is not defined ?

Options

A. `sin^(-1)x`
B. `tan^(-1)x`
C. `sec^(-1)x`
D. `cos^(-1)x`

Answer

To solve the problem, we need to analyze the given condition and determine which of the inverse trigonometric functions is not defined for the specified range of \( |x| \).

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We are given that \( \frac{1}{2} < |x| < 1 \). This means that \( x \) can take values in the intervals:
- \( -1 < x < -\frac{1}{2} \)
- \( \frac{1}{2} < x < 1 \)

2. **Identify the Inverse Trigonometric Functions**:
We need to check the following functions:
- \( \sin^{-1}(x) \)
- \( \tan^{-1}(x) \)
- \( \sec^{-1}(x) \)
- \( \cos^{-1}(x) \)

3. **Check the Domain of Each Function**:
- **For \( \sin^{-1}(x) \)**:
The domain is \( -1 \leq x \leq 1 \). Since our intervals \( (-1, -\frac{1}{2}) \) and \( (\frac{1}{2}, 1) \) are within this range, \( \sin^{-1}(x) \) is defined.

- **For \( \tan^{-1}(x) \)**:
The domain is all real numbers \( (-\infty, \infty) \). Therefore, \( \tan^{-1}(x) \) is defined for all \( x \) in the given intervals.

- **For \( \sec^{-1}(x) \)**:
The domain is \( |x| \geq 1 \) or \( x \leq -1 \) or \( x \geq 1 \). Since \( |x| \) is between \( \frac{1}{2} \) and \( 1 \), \( \sec^{-1}(x) \) is not defined in this range.

- **For \( \cos^{-1}(x) \)**:
The domain is \( -1 \leq x \leq 1 \). Similar to \( \sin^{-1}(x) \), the intervals \( (-1, -\frac{1}{2}) \) and \( (\frac{1}{2}, 1) \) are within this range, so \( \cos^{-1}(x) \) is defined.

4. **Conclusion**:
The only function that is not defined in the given range \( \frac{1}{2} < |x| < 1 \) is \( \sec^{-1}(x) \).

### Final Answer:
The function that is not defined is \( \sec^{-1}(x) \).