Simplify (x-y)/(x^2-1)*(x-1)/(x^2-y^2)

$\frac{x - y}{x^{2} - 1} \cdot \frac{x - 1}{x^{2} - y^{2}}$

Step 1

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{x - y}{x^{2} - 1^{2}} \cdot \frac{x - 1}{x^{2} - y^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{x - y}{(x + 1) (x - 1)} \cdot \frac{x - 1}{x^{2} - y^{2}}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$\frac{x - y}{(x + 1) (x - 1)} \cdot \frac{x - 1}{(x + y) (x - y)}$

Step 3

Simplify terms.

Cancel the common factor of $x - y$ .

Factor $x - y$ out of $(x + y) (x - y)$ .

$\frac{x - y}{(x + 1) (x - 1)} \cdot \frac{x - 1}{(x - y) (x + y)}$

Cancel the common factor.

$\frac{x - y}{(x + 1) (x - 1)} \cdot \frac{x - 1}{(x - y) (x + y)}$

Rewrite the expression.

$\frac{1}{(x + 1) (x - 1)} \cdot \frac{x - 1}{x + y}$

Cancel the common factor of $x - 1$ .

Factor $x - 1$ out of $(x + 1) (x - 1)$ .

$\frac{1}{(x - 1) (x + 1)} \cdot \frac{x - 1}{x + y}$

Cancel the common factor.

$\frac{1}{(x - 1) (x + 1)} \cdot \frac{x - 1}{x + y}$

Rewrite the expression.

$\frac{1}{x + 1} \cdot \frac{1}{x + y}$

Multiply $\frac{1}{x + 1}$ by $\frac{1}{x + y}$ .

$\frac{1}{(x + 1) (x + y)}$