Statement -1 : If ` x in R, x ne 0` , then `x^2 + 1/(x^(2))` cannot be equal to `cos theta` for any `theta`
Statement -2 : Sum of a positive number and its reciprocal cannot be less than 2.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the given problem, we need to analyze both statements and determine their validity.

### Step 1: Analyze Statement 1
**Statement 1**: If \( x \in \mathbb{R}, x \neq 0 \), then \( x^2 + \frac{1}{x^2} \) cannot be equal to \( \cos \theta \) for any \( \theta \).

1. **Determine the range of \( \cos \theta \)**:
- The range of \( \cos \theta \) is \([-1, 1]\).

2. **Analyze the expression \( x^2 + \frac{1}{x^2} \)**:
- Let \( y = x^2 \). Since \( x \neq 0 \), \( y > 0 \).
- We can rewrite the expression as \( y + \frac{1}{y} \).

3. **Find the minimum value of \( y + \frac{1}{y} \)**:
- By the AM-GM inequality, we have:
\[ y + \frac{1}{y} \geq 2 \]
- The equality holds when \( y = 1 \) (i.e., \( x = 1 \) or \( x = -1 \)).
- Thus, \( x^2 + \frac{1}{x^2} \geq 2 \).

4. **Conclusion for Statement 1**:
- Since \( x^2 + \frac{1}{x^2} \) is always greater than or equal to 2, and the maximum value of \( \cos \theta \) is 1, it follows that:
\[ x^2 + \frac{1}{x^2} \neq \cos \theta \quad \text{for any } \theta. \]
- Therefore, **Statement 1 is true**.

### Step 2: Analyze Statement 2
**Statement 2**: The sum of a positive number and its reciprocal cannot be less than 2.

1. **Let \( z \) be a positive number**:
- We need to show \( z + \frac{1}{z} \geq 2 \).

2. **Using the AM-GM inequality**:
- By the AM-GM inequality:
\[ z + \frac{1}{z} \geq 2\sqrt{z \cdot \frac{1}{z}} = 2. \]
- The equality holds when \( z = 1 \).

3. **Conclusion for Statement 2**:
- Thus, the statement is also true: the sum of a positive number and its reciprocal is always greater than or equal to 2.

### Final Conclusion
- Both statements are true.
- Statement 2 correctly explains Statement 1 because the minimum value of \( x^2 + \frac{1}{x^2} \) being 2 supports the conclusion that it cannot equal \( \cos \theta \) which is at most 1.

### Answer
Both statements are true, and Statement 2 is a correct explanation for Statement 1.