`((1)/(x) + (1)/(y)) ((1)/(x) - (1)/(y)) = ?`

Options

A. `((1)/(x^(2)) - (1)/(y^(2)))`
B. `((1)/(x^(2)) + (1)/(y^(2)))`
C. `((1)/(x^(2)) + (1)/(y^(2)) - (1)/(xy))`
D. `((1)/(x^(2)) - (1)/(y^(2)) + (1)/(xy))`

Answer

To solve the expression \(\left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x} - \frac{1}{y}\right)\), we can use the identity for the difference of squares, which states that:

\[ (a + b)(a - b) = a^2 - b^2 \]

### Step-by-step Solution:

1. **Identify \(a\) and \(b\)**:
- Here, let \(a = \frac{1}{x}\) and \(b = \frac{1}{y}\).

2. **Apply the identity**:
- According to the difference of squares identity, we have:
\[ \left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x} - \frac{1}{y}\right) = \left(\frac{1}{x}\right)^2 - \left(\frac{1}{y}\right)^2 \]

3. **Calculate \(\left(\frac{1}{x}\right)^2\) and \(\left(\frac{1}{y}\right)^2\)**:
- \(\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}\)
- \(\left(\frac{1}{y}\right)^2 = \frac{1}{y^2}\)

4. **Substitute back into the equation**:
- Now substituting these values back, we get:
\[ \frac{1}{x^2} - \frac{1}{y^2} \]

5. **Final Result**:
- Therefore, the final answer is:
\[ \frac{1}{x^2} - \frac{1}{y^2} \]