Which one of the following statement is meaningless ?

Options

A. `cos^(-1) ( " In " ((2e + 4)/3))`
B. `cosec^(-1) (pi/3)`
C. `cot^(_1) ( pi/2)`
D. `sec^(-1) (pi)`

Answer

To determine which statement is meaningless among the given options, we need to analyze each option based on the properties and ranges of inverse trigonometric functions.

### Step-by-Step Solution:

1. **Understanding the Range of Cosine Inverse Function**:
- The range of \( \cos^{-1}(x) \) is defined as \( [0, \pi] \) for \( x \) in the domain \( [-1, 1] \). Thus, any input to \( \cos^{-1}(x) \) must be within this range.

2. **Analyzing Option A: \( \cos^{-1}(\ln(2e) + \frac{4}{3}) \)**:
- First, we calculate \( \ln(2e) + \frac{4}{3} \):
\[ \ln(2e) = \ln(2) + \ln(e) = \ln(2) + 1 \]
Therefore,
\[ \ln(2e) + \frac{4}{3} = \ln(2) + 1 + \frac{4}{3} \]
- Since \( \ln(2) \) is approximately \( 0.693 \), we can estimate:
\[ \ln(2) + 1 \approx 1.693 \]
Adding \( \frac{4}{3} \) (which is approximately \( 1.333 \)):
\[ 1.693 + 1.333 \approx 3.026 \]
- Since \( 3.026 > 1 \), this value is outside the valid range for \( \cos^{-1}(x) \). Thus, \( \cos^{-1}(\ln(2e) + \frac{4}{3}) \) is meaningless.

3. **Analyzing Option B: \( \cos^{-1}(\frac{\pi}{3}) \)**:
- The value \( \frac{\pi}{3} \) is approximately \( 1.047 \), which is within the range \( [-1, 1] \). Thus, \( \cos^{-1}(\frac{\pi}{3}) \) is valid.

4. **Analyzing Option C: \( \cot^{-1}(\frac{\pi}{2}) \)**:
- The value \( \cot^{-1}(x) \) is defined for all real \( x \). Therefore, \( \cot^{-1}(\frac{\pi}{2}) \) is also valid.

5. **Analyzing Option D: \( \sec^{-1}(\pi) \)**:
- The value \( \sec^{-1}(x) \) is defined for \( |x| \geq 1 \). Since \( \pi \approx 3.14 \), this is valid as well.

### Conclusion:
- The only meaningless statement is **Option A: \( \cos^{-1}(\ln(2e) + \frac{4}{3}) \)**.