The value of `(x + y)/(x - y) + (x - y)/(x + y) - (2(x^2 - y^2))/(x^2 - y^2)` is :

Options

A. `1`
B. `x`
C. `y`
D. `0`

Answer

To solve the expression \(\frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2(x^2 - y^2)}{x^2 - y^2}\), we will follow these steps:

### Step 1: Identify the common denominator
The denominators in the first two fractions are \(x - y\) and \(x + y\). The least common denominator (LCD) for these fractions is \( (x^2 - y^2) \), since \( x^2 - y^2 = (x - y)(x + y) \).

### Step 2: Rewrite each term with the common denominator
We can rewrite the expression as follows:

\[ \frac{(x + y)^2}{(x - y)(x + y)} + \frac{(x - y)^2}{(x + y)(x - y)} - \frac{2(x^2 - y^2)}{x^2 - y^2} \]

This simplifies to:

\[ \frac{(x + y)^2 + (x - y)^2 - 2(x^2 - y^2)}{x^2 - y^2} \]

### Step 3: Expand the numerators
Now we will expand the numerators:

1. \((x + y)^2 = x^2 + 2xy + y^2\)
2. \((x - y)^2 = x^2 - 2xy + y^2\)

Putting these into the expression gives:

\[ \frac{(x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) - 2(x^2 - y^2)}{x^2 - y^2} \]

### Step 4: Combine like terms in the numerator
Now we combine the terms in the numerator:

\[ = \frac{(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) - 2(x^2 - y^2)}{x^2 - y^2} \]

Simplifying the numerator:

\[ = \frac{2x^2 + 2y^2 - 2(x^2 - y^2)}{x^2 - y^2} \]

### Step 5: Simplify the numerator further
Distributing the \(-2\) in the numerator:

\[ = \frac{2x^2 + 2y^2 - 2x^2 + 2y^2}{x^2 - y^2} \]

This simplifies to:

\[ = \frac{4y^2}{x^2 - y^2} \]

### Step 6: Final expression
Thus, the value of the original expression is:

\[ \frac{4y^2}{x^2 - y^2} \]