Simplify (x-1/x)/(x+1/x)

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

Step 1

Multiply the numerator and denominator of the complex fraction by $x$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Raise $x$ to the power of $1$ .

$\frac{x^{1} x - 1}{x \cdot x + 1}$

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} - 1}{x \cdot x + 1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 1}{x \cdot x + 1}$

Add $1$ and $1$ .

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

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

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

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 $b = 1$ .

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

Step 5

Multiply $x$ by $x$ .

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