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

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

Step 1

Simplify the numerator.

Factor $x^{- 1}$ out of $x - x^{- 1}$ .

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

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

Factor $x^{- 1}$ out of $x^{1}$ .

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

Factor $x^{- 1}$ out of $- x^{- 1}$ .

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

Factor $x^{- 1}$ out of $x^{- 1} x^{2} + x^{- 1} \cdot - 1$ .

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

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

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

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 $b = 1$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Factor $x^{- 1}$ out of $x + x^{- 1}$ .

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

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

Factor $x^{- 1}$ out of $x^{1}$ .

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

Multiply by $1$ .

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

Factor $x^{- 1}$ out of $x^{- 1} x^{2} + x^{- 1} \cdot 1$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Cancel the common factor of $\frac{1}{x}$ .

Cancel the common factor.

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

Rewrite the expression.

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