Trigonometry Examples

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

Step 1

Subtract $\frac{1}{x}$ from both sides of the equation.

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, x$

Step 2.2

Since $y, 1, x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $y^{1}, x^{1}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 2.7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 2.8

The LCM of $y^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot x$

Step 2.9

Multiply $y$ by $x$ .

$y x$

Step 3

Multiply each term in $\frac{1}{y} = 1 - \frac{1}{x}$ by $y x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{y} = 1 - \frac{1}{x}$ by $y x$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y$ .

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Step 3.2.1.1

Factor $y$ out of $y x$ .

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $y x$ by $1$ .

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

Step 3.3.1.2

Cancel the common factor of $x$ .

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

Factor $x$ out of $y x$ .

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

Step 3.3.1.2.3

Cancel the common factor.

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

Step 3.3.1.2.4

Rewrite the expression.

$x = y x - y$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $y x - y = x$ .

$y x - y = x$

Step 4.2

Factor $y$ out of $y x - y$ .

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Step 4.2.1

Factor $y$ out of $y x$ .

$y (x) - y = x$

Step 4.2.2

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = x$

Step 4.2.3

Factor $y$ out of $y (x) + y \cdot - 1$ .

$y (x - 1) = x$

Step 4.3

Divide each term in $y (x - 1) = x$ by $x - 1$ and simplify.

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Step 4.3.1

Divide each term in $y (x - 1) = x$ by $x - 1$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x - 1$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $y$ by $1$ .

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

Step 5

Interchange the variables.

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

Step 6

Solve for $y$ .

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Step 6.1

Multiply the equation by $y - 1$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify $(y - 1) (x)$ .

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Step 6.2.1.1

Apply the distributive property.

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

Step 6.2.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $y - 1$ .

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Step 6.3.1.1

Cancel the common factor.

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

Step 6.3.1.2

Rewrite the expression.

$y x - x = y$

Step 6.4

Solve for $y$ .

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Step 6.4.1

Subtract $y$ from both sides of the equation.

$y x - x - y = 0$

Step 6.4.2

Add $x$ to both sides of the equation.

$y x - y = x$

Step 6.4.3

Factor $y$ out of $y x - y$ .

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Step 6.4.3.1

Factor $y$ out of $y x$ .

$y (x) - y = x$

Step 6.4.3.2

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = x$

Step 6.4.3.3

Factor $y$ out of $y (x) + y \cdot - 1$ .

$y (x - 1) = x$

Step 6.4.4

Divide each term in $y (x - 1) = x$ by $x - 1$ and simplify.

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Step 6.4.4.1

Divide each term in $y (x - 1) = x$ by $x - 1$ .

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

Step 6.4.4.2

Simplify the left side.

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Step 6.4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 6.4.4.2.1.1

Cancel the common factor.

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

Step 6.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 7

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{x}{x - 1}$

Step 8

Verify if $f^{- 1} (x) = \frac{x}{x - 1}$ is the inverse of $f (x) = \frac{x}{x - 1}$ .

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Step 8.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 8.2

Evaluate $f^{- 1} (f (x))$ .

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Step 8.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 8.2.2

Evaluate $f^{- 1} (\frac{x}{x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{x - 1}) = \frac{\frac{x}{x - 1}}{(\frac{x}{x - 1}) - 1}$

Step 8.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.4

Simplify the denominator.

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Step 8.2.4.1

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

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

Step 8.2.4.2

Combine $- 1$ and $\frac{x - 1}{x - 1}$ .

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

Step 8.2.4.3

Combine the numerators over the common denominator.

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

Step 8.2.4.4

Rewrite $\frac{x - (x - 1)}{x - 1}$ in a factored form.

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Step 8.2.4.4.1

Apply the distributive property.

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

Step 8.2.4.4.2

Multiply $- 1$ by $- 1$ .

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

Step 8.2.4.4.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{0 + 1}{x - 1}}$

Step 8.2.4.4.4

Add $0$ and $1$ .

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

Step 8.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.6

Cancel the common factor of $x - 1$ .

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Step 8.2.6.1

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

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

Step 8.2.6.2

Cancel the common factor.

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

Step 8.2.6.3

Rewrite the expression.

$f^{- 1} (\frac{x}{x - 1}) = x$

Step 8.3

Evaluate $f (f^{- 1} (x))$ .

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Step 8.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 8.3.2

Evaluate $f (\frac{x}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{x - 1}) = \frac{\frac{x}{x - 1}}{(\frac{x}{x - 1}) - 1}$

Step 8.3.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{x}{x - 1} - 1}$

Step 8.3.4

Simplify the denominator.

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Step 8.3.4.1

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

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{x}{x - 1} - 1 \cdot \frac{x - 1}{x - 1}}$

Step 8.3.4.2

Combine $- 1$ and $\frac{x - 1}{x - 1}$ .

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

Step 8.3.4.3

Combine the numerators over the common denominator.

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{x - (x - 1)}{x - 1}}$

Step 8.3.4.4

Rewrite $\frac{x - (x - 1)}{x - 1}$ in a factored form.

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Step 8.3.4.4.1

Apply the distributive property.

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

Step 8.3.4.4.2

Multiply $- 1$ by $- 1$ .

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

Step 8.3.4.4.3

Subtract $x$ from $x$ .

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{0 + 1}{x - 1}}$

Step 8.3.4.4.4

Add $0$ and $1$ .

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{1}{x - 1}}$

Step 8.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot (1 (x - 1))$

Step 8.3.6

Cancel the common factor of $x - 1$ .

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Step 8.3.6.1

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

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot ((x - 1) \cdot 1)$

Step 8.3.6.2

Cancel the common factor.

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot ((x - 1) \cdot 1)$

Step 8.3.6.3

Rewrite the expression.

$f (\frac{x}{x - 1}) = x$

Step 8.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{x - 1}$ is the inverse of $f (x) = \frac{x}{x - 1}$ .

$f^{- 1} (x) = \frac{x}{x - 1}$