Simplify (1/((x-1)^2))÷(x/(x^2-1))

$\frac{1}{{(x - 1)}^{2}} \div \frac{x}{x^{2} - 1}$

To divide by a fraction, multiply by its reciprocal.

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

Simplify the numerator.

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

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

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 $b = 1$ .

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

Simplify terms.

Combine.

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

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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

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

Cancel the common factors.

Factor $x - 1$ out of ${(x - 1)}^{2} x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Multiply $x + 1$ by $1$ .

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

Reorder factors in $\frac{x + 1}{(x - 1) x}$ .

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