Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1 - \cot (- y)}{1 + \cot (y)} = x$ .

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

Step 2.2

Multiply both sides by $1 + \cot (y)$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{1 - \cot (- y)}{1 + \cot (y)} (1 + \cot (y))$ .

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Step 2.3.1.1.1

Cancel the common factor of $1 + \cot (y)$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

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

Step 2.3.1.1.1.2

Rewrite the expression.

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

Step 2.3.1.1.2

Simplify each term.

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Step 2.3.1.1.2.1

Since $\cot (- y)$ is an odd function, rewrite $\cot (- y)$ as $- \cot (y)$ .

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

Step 2.3.1.1.2.2

Multiply $- - \cot (y)$ .

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Step 2.3.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 + 1 \cot (y) = x (1 + \cot (y))$

Step 2.3.1.1.2.2.2

Multiply $\cot (y)$ by $1$ .

$1 + \cot (y) = x (1 + \cot (y))$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $x (1 + \cot (y))$ .

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Step 2.3.2.1.1

Apply the distributive property.

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

Step 2.3.2.1.2

Multiply $x$ by $1$ .

$1 + \cot (y) = x + x \cot (y)$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Substitute $u$ for $\cot (y)$ .

$1 + u = x + x (u)$

Step 2.4.2

Subtract $x u$ from both sides of the equation.

$1 + u - x u = x$

Step 2.4.3

Subtract $1$ from both sides of the equation.

$u - x u = x - 1$

Step 2.4.4

Factor $u$ out of $u - x u$ .

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Step 2.4.4.1

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - x u = x - 1$

Step 2.4.4.2

Factor $u$ out of $- x u$ .

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

Step 2.4.4.3

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

$u (1 - x) = x - 1$

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

Simplify the left side.

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Step 2.4.5.2.1

Cancel the common factor of $1 - x$ .

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Step 2.4.5.2.1.1

Cancel the common factor.

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

Step 2.4.5.2.1.2

Divide $u$ by $1$ .

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

Step 2.4.5.3

Simplify the right side.

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Step 2.4.5.3.1

Combine the numerators over the common denominator.

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

Step 2.4.5.3.2

Cancel the common factor of $x - 1$ and $1 - x$ .

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Step 2.4.5.3.2.1

Factor $- 1$ out of $x$ .

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

Step 2.4.5.3.2.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.4.5.3.2.3

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

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

Step 2.4.5.3.2.4

Reorder terms.

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

Step 2.4.5.3.2.5

Cancel the common factor.

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

Step 2.4.5.3.2.6

Divide $- 1$ by $1$ .

$u = - 1$

Step 2.4.6

Substitute $\cot (y)$ for $u$ .

$\cot (y) = - 1$

Step 2.4.7

Take the inverse cotangent of both sides of the equation to extract $y$ from inside the cotangent.

$y = arccot (- 1)$

Step 2.4.8

Simplify the right side.

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Step 2.4.8.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$y = \frac{3 π}{4}$

Step 2.4.9

The cotangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$y = \frac{3 π}{4} - π$

Step 2.4.10

Simplify the expression to find the second solution.

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Step 2.4.10.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$y = \frac{3 π}{4} - π + 2 π$

Step 2.4.10.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$y = \frac{7 π}{4}$

Step 2.4.11

Find the period of $\cot (y)$ .

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Step 2.4.11.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 2.4.11.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 2.4.11.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 2.4.11.4

Divide $π$ by $1$ .

$π$

Step 2.4.12

The period of the $\cot (y)$ function is $π$ so values will repeat every $π$ radians in both directions.

$y = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 2.5

Consolidate the answers.

$y = \frac{3 π}{4} + π n$ , for any integer $n$

Step 3

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

$f^{- 1} (x) = \frac{3 π}{4} + π n$

Step 4

Verify if $f^{- 1} (x) = \frac{3 π}{4} + π n$ is the inverse of $f (x) = \frac{1 - \cot (- x)}{1 + \cot (x)}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (\frac{1 - \cot (- x)}{1 + \cot (x)}) = \frac{3 π}{4} + π n$

Step 4.2.3

Reorder $\frac{3 π}{4}$ and $π n$ .

$f^{- 1} (\frac{1 - \cot (- x)}{1 + \cot (x)}) = π n + \frac{3 π}{4}$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{3 π}{4} + π n)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 π}{4} + π n) = \frac{1 - \cot (- (\frac{3 π}{4} + π n))}{1 + \cot (\frac{3 π}{4} + π n)}$

Step 4.3.3

Apply the distributive property.

$f (\frac{3 π}{4} + π n) = \frac{1 - \cot (- \frac{3 π}{4} - π n)}{1 + \cot (\frac{3 π}{4} + π n)}$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 π}{4} + π n$ is the inverse of $f (x) = \frac{1 - \cot (- x)}{1 + \cot (x)}$ .

$f^{- 1} (x) = \frac{3 π}{4} + π n$