Select the wrong option

Options

A. `-1 le sin^(-1)x le 1 rArr -sin1 le x le sin1`
B. `pi/3 le cos^(-1)x le (4pi)/3 rArr-1/2 le x le 1/2`
C. `pi/4 le cot^(-1)x le (5pi)/6 rArr -sqrt3 le x le 1`
D. `sec^(-1)x ge pi/4rArr x le sqrt2`

Answer

To solve the problem of identifying the wrong option regarding the ranges of inverse trigonometric functions, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Option 1:** The range of \( \sin^{-1} x \) is from \(-1\) to \(1\).

**Solution:**
The range of \( \sin^{-1} x \) (also known as arcsine) is actually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). However, the values of \(x\) for which \( \sin^{-1} x \) is defined are from \(-1\) to \(1\). Thus, the statement about the range of \( \sin^{-1} x \) being from \(-1\) to \(1\) is misleading, as it refers to the domain of \(x\) rather than the range of the function itself.

**Conclusion:** This option is somewhat misleading but not outright wrong in the context of the domain.

### Step 2: Analyze Option 2
**Option 2:** The range of \( \cos^{-1} x \) is from \( \frac{\pi}{3} \) to \( \frac{4\pi}{3} \).

**Solution:**
The range of \( \cos^{-1} x \) (arccosine) is from \(0\) to \(\pi\). Since \( \frac{4\pi}{3} \) is greater than \(\pi\), this statement is incorrect.

**Conclusion:** This option is wrong.

### Step 3: Analyze Option 3
**Option 3:** The range of \( \cot^{-1} x \) is from \( \frac{\pi}{4} \) to \( \frac{5\pi}{6} \).

**Solution:**
The range of \( \cot^{-1} x \) is actually from \(0\) to \(\pi\). Both \( \frac{\pi}{4} \) and \( \frac{5\pi}{6} \) fall within this range, making this statement correct.

**Conclusion:** This option is correct.

### Step 4: Analyze Option 4
**Option 4:** \( \sec^{-1} x \) is greater than or equal to \( \frac{\pi}{4} \), which means \( x \) is greater than or equal to \( \sec(\frac{\pi}{4}) \).

**Solution:**
The value of \( \sec(\frac{\pi}{4}) \) is \( \sqrt{2} \). Therefore, the statement implies that \( x \geq \sqrt{2} \). However, if the option states that \( x \) is less than \( \sqrt{2} \), this contradicts the earlier statement. Thus, this statement is also incorrect.

**Conclusion:** This option is wrong.

### Final Conclusion
The wrong options are:
- Option 2: The range of \( \cos^{-1} x \) is from \( \frac{\pi}{3} \) to \( \frac{4\pi}{3} \) (incorrect).
- Option 4: \( \sec^{-1} x \) is greater than or equal to \( \frac{\pi}{4} \) while stating \( x \) is less than \( \sqrt{2} \) (incorrect).