Statement I `cosec^(-1)(1/2 +1/sqrt2) gt sec^(-1)(1/2+1/sqrt2)`
Statement II `cosec^(-1) x gt sec^(-1) x", if " 1 le x lt sqrt2`

Options

A. Statement I is True, Statement II is True, Statement II is a correct explanation for statement I
B. Statement I is True, Statement II is True, Statement II is NOT a correct explanation for Statement I
C. Statement I is True, Statement II is False
D. Statement I is False, Statement II is True.

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Understand the Statements

**Statement I:**
\[ \csc^{-1}\left(\frac{1}{2} + \frac{1}{\sqrt{2}}\right) > \sec^{-1}\left(\frac{1}{2} + \frac{1}{\sqrt{2}}\right) \]

**Statement II:**
\[ \csc^{-1}(x) > \sec^{-1}(x) \quad \text{for } 1 \leq x < \sqrt{2} \]

### Step 2: Evaluate the Expression \(\frac{1}{2} + \frac{1}{\sqrt{2}}\)

First, we calculate the value of \(x\):
\[ x = \frac{1}{2} + \frac{1}{\sqrt{2}} = \frac{1}{2} + \frac{\sqrt{2}}{2} = \frac{1 + \sqrt{2}}{2} \]

### Step 3: Check the Range of \(x\)

Next, we need to check if \(x\) lies in the interval \([1, \sqrt{2})\).

1. **Check if \(x \geq 1\)**:
\[ \frac{1 + \sqrt{2}}{2} \geq 1 \implies 1 + \sqrt{2} \geq 2 \implies \sqrt{2} \geq 1 \quad \text{(True)} \]

2. **Check if \(x < \sqrt{2}\)**:
\[ \frac{1 + \sqrt{2}}{2} < \sqrt{2} \implies 1 + \sqrt{2} < 2\sqrt{2} \implies 1 < \sqrt{2} \quad \text{(True)} \]

Thus, \(x = \frac{1 + \sqrt{2}}{2}\) lies in the interval \([1, \sqrt{2})\).

### Step 4: Analyze the Functions \(\csc^{-1}(x)\) and \(\sec^{-1}(x)\)

- The function \(\csc^{-1}(x)\) is defined for \(x \geq 1\) and is a decreasing function.
- The function \(\sec^{-1}(x)\) is defined for \(x \geq 1\) and is an increasing function.

### Step 5: Compare \(\csc^{-1}(x)\) and \(\sec^{-1}(x)\)

Since \(\csc^{-1}(x)\) is decreasing and \(\sec^{-1}(x)\) is increasing, for \(1 \leq x < \sqrt{2}\), it follows that:
\[ \csc^{-1}(x) > \sec^{-1}(x) \]

### Step 6: Conclusion on Statement II

Since \(x = \frac{1 + \sqrt{2}}{2}\) lies in the interval \([1, \sqrt{2})\), we conclude that:
\[ \csc^{-1}\left(\frac{1}{2} + \frac{1}{\sqrt{2}}\right) > \sec^{-1}\left(\frac{1}{2} + \frac{1}{\sqrt{2}}\right) \]

Thus, **Statement I is true** and **Statement II correctly explains Statement I**.

### Final Answer

Both statements are true, and Statement II correctly explains Statement I.

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