Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 1.4

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.5

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Reorder the factors of $y x$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 2.4

To write $- \frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.5

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Reorder the factors of $y x$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $x y$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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