Consider the following statements:
I. `n(sin^2 67(1^@)/(2)-sin^2 22 (1^@)/(2)) gt 1` for all positive integers `n ge 2`.
II. If x is any positive real number, then `nx gt 1` for all positive integers `n ge 2`.
Which of the above statements is/are correct?

Options

A. A) Only I
B. B) Only II
C. C) Both I and II
D. D) Neither I nor II

Answer

To solve the question, we need to analyze the two statements provided and determine their validity.

**Statement I:**
We need to verify if \( n(\sin^2(67.5^\circ) - \sin^2(22.5^\circ)) > 1 \) for all positive integers \( n \geq 2 \).

1. **Calculate \( \sin^2(67.5^\circ) \) and \( \sin^2(22.5^\circ) \)**:
- We can use the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \).
- For \( \sin^2(67.5^\circ) \):
\[ \sin^2(67.5^\circ) = \frac{1 - \cos(135^\circ)}{2} = \frac{1 - (-\frac{1}{\sqrt{2}})}{2} = \frac{1 + \frac{1}{\sqrt{2}}}{2} = \frac{2 + \sqrt{2}}{4} \]
- For \( \sin^2(22.5^\circ) \):
\[ \sin^2(22.5^\circ) = \frac{1 - \cos(45^\circ)}{2} = \frac{1 - \frac{1}{\sqrt{2}}}{2} = \frac{2 - \sqrt{2}}{4} \]

2. **Subtract the two values**:
\[ \sin^2(67.5^\circ) - \sin^2(22.5^\circ) = \frac{2 + \sqrt{2}}{4} - \frac{2 - \sqrt{2}}{4} = \frac{(2 + \sqrt{2}) - (2 - \sqrt{2})}{4} = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2} \]

3. **Multiply by \( n \)**:
\[ n(\sin^2(67.5^\circ) - \sin^2(22.5^\circ)) = n \cdot \frac{\sqrt{2}}{2} \]

4. **Check if \( n \cdot \frac{\sqrt{2}}{2} > 1 \)** for \( n \geq 2 \):
- For \( n = 2 \):
\[ 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \approx 1.414 > 1 \]
- For \( n = 3 \):
\[ 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \approx 2.121 > 1 \]
- This holds true for all \( n \geq 2 \).

**Conclusion for Statement I**: True.

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**Statement II:**
We need to verify if \( nx > 1 \) for all positive integers \( n \geq 2 \) and any positive real number \( x \).

1. **Choose a counterexample**:
- Let \( n = 2 \) and \( x = \frac{1}{4} \):
\[ nx = 2 \cdot \frac{1}{4} = \frac{1}{2} < 1 \]

**Conclusion for Statement II**: False.

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**Final Conclusion**:
- Statement I is correct.
- Statement II is incorrect.

Thus, the correct answer is that only Statement I is true.