The simplified form of the rational expression `((x^(2))/(x^(2)-y^(2))-1)((x-y)/y+2)` is

Options

A. `x/(x+y)`
B. `y/(x+y)`
C. `y/(x-y)`
D. `x/(x-y)`

Answer

To simplify the expression \(\left(\frac{x^2}{x^2 - y^2} - 1\right)\left(\frac{x - y}{y} + 2\right)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\frac{x^2}{x^2 - y^2} - 1\right)\left(\frac{x - y}{y} + 2\right) \]

### Step 2: Find a common denominator for the first part
The first part is \(\frac{x^2}{x^2 - y^2} - 1\). To combine these terms, we need a common denominator:
\[ \frac{x^2}{x^2 - y^2} - \frac{x^2 - y^2}{x^2 - y^2} = \frac{x^2 - (x^2 - y^2)}{x^2 - y^2} \]
This simplifies to:
\[ \frac{y^2}{x^2 - y^2} \]

### Step 3: Simplify the second part
Now, we simplify the second part \(\frac{x - y}{y} + 2\):
\[ \frac{x - y}{y} + 2 = \frac{x - y}{y} + \frac{2y}{y} = \frac{x - y + 2y}{y} = \frac{x + y}{y} \]

### Step 4: Combine the two parts
Now we can combine the two simplified parts:
\[ \frac{y^2}{x^2 - y^2} \cdot \frac{x + y}{y} \]

### Step 5: Multiply the fractions
Multiplying the fractions gives us:
\[ \frac{y^2(x + y)}{y(x^2 - y^2)} \]

### Step 6: Simplify the expression
We can cancel \(y\) in the numerator and denominator:
\[ \frac{y(x + y)}{x^2 - y^2} \]

### Step 7: Factor the denominator
Recall that \(x^2 - y^2\) can be factored using the difference of squares:
\[ x^2 - y^2 = (x + y)(x - y) \]

### Step 8: Final simplification
Now substituting this back into our expression:
\[ \frac{y(x + y)}{(x + y)(x - y)} \]
We can cancel \(x + y\) from the numerator and denominator (assuming \(x + y \neq 0\)):
\[ \frac{y}{x - y} \]

### Final Answer
Thus, the simplified form of the given rational expression is:
\[ \frac{y}{x - y} \]