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

$y$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.1.1

Simplify

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

Tap for more steps...

Step 2.3.1.1.1

Cancel the common factor of

$1 + \cot (y)$ .

Tap for more steps...

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.

Tap for more steps...

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

$- - \cot (y)$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.3.2.1

Simplify

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

Tap for more steps...

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

$y$ .

Tap for more steps...

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

$u - x u$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 2.4.5.2.1

Cancel the common factor of

$1 - x$ .

Tap for more steps...

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.

Tap for more steps...

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

$1 - x$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$\cot (y)$ .

Tap for more steps...

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

$n$

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

$n$

Step 2.5

Consolidate the answers.

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

$n$

$y = \frac{3 π}{4} + π n$ , 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

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

Tap for more steps...

Step 4.1

To verify the inverse, check if

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

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

Step 4.2

Evaluate

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

Tap for more steps...

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

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

Tap for more steps...

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

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

$f^{- 1} (x) = \frac{3 π}{4} + π n$