Trigonometry Examples

$\frac{\cos (2 x)}{\sin (x)}$

Step 1

Interchange the variables.

$x = \frac{\cos (2 y)}{\sin (y)}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{\cos (2 y)}{\sin (y)} = x$ .

$\frac{\cos (2 y)}{\sin (y)} = x$

Step 2.2

Multiply both sides by $\sin (y)$ .

$\frac{\cos (2 y)}{\sin (y)} \sin (y) = x \sin (y)$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $\sin (y)$ .

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Step 2.3.1.1

Cancel the common factor.

$\frac{\cos (2 y)}{\sin (y)} \sin (y) = x \sin (y)$

Step 2.3.1.2

Rewrite the expression.

$\cos (2 y) = x \sin (y)$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$1 - 2 \sin^{2} (y) = x \sin (y)$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$- 2 \sin^{2} (y) = x \sin (y) - 1$

Step 2.4.3

Subtract $x \sin (y)$ from both sides of the equation.

$- 2 \sin^{2} (y) - x \sin (y) = - 1$

Step 2.4.4

Solve the equation for $y$ .

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Step 2.4.4.1

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

$- 2 {(u)}^{2} - x u = - 1$

Step 2.4.4.2

Add $1$ to both sides of the equation.

$- 2 u^{2} - x u + 1 = 0$

Step 2.4.4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.4.4

Substitute the values $a = - 2$ , $b = - x$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{x \pm \sqrt{{(- x)}^{2} - 4 \cdot (- 2 \cdot 1)}}{2 \cdot - 2}$

Step 2.4.4.5

Simplify.

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Step 2.4.4.5.1

Simplify the numerator.

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Step 2.4.4.5.1.1

Apply the product rule to $- x$ .

$u = \frac{x \pm \sqrt{{(- 1)}^{2} x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.5.1.2

Raise $- 1$ to the power of $2$ .

$u = \frac{x \pm \sqrt{1 x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.5.1.3

Multiply $x^{2}$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.5.1.4

Multiply $- 4 \cdot - 2 \cdot 1$ .

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Step 2.4.4.5.1.4.1

Multiply $- 4$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.5.1.4.2

Multiply $8$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{2 \cdot - 2}$

Step 2.4.4.5.2

Multiply $2$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{- 4}$

Step 2.4.4.5.3

Move the negative in front of the fraction.

$u = - \frac{x \pm \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.4.6.1

Simplify the numerator.

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Step 2.4.4.6.1.1

Apply the product rule to $- x$ .

$u = \frac{x \pm \sqrt{{(- 1)}^{2} x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.6.1.2

Raise $- 1$ to the power of $2$ .

$u = \frac{x \pm \sqrt{1 x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.6.1.3

Multiply $x^{2}$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.6.1.4

Multiply $- 4 \cdot - 2 \cdot 1$ .

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Step 2.4.4.6.1.4.1

Multiply $- 4$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.6.1.4.2

Multiply $8$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{2 \cdot - 2}$

Step 2.4.4.6.2

Multiply $2$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{- 4}$

Step 2.4.4.6.3

Move the negative in front of the fraction.

$u = - \frac{x \pm \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.6.4

Change the $\pm$ to $+$ .

$u = - \frac{x + \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.4.7.1

Simplify the numerator.

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Step 2.4.4.7.1.1

Apply the product rule to $- x$ .

$u = \frac{x \pm \sqrt{{(- 1)}^{2} x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.7.1.2

Raise $- 1$ to the power of $2$ .

$u = \frac{x \pm \sqrt{1 x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.7.1.3

Multiply $x^{2}$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} - 4 \cdot - 2 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.7.1.4

Multiply $- 4 \cdot - 2 \cdot 1$ .

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Step 2.4.4.7.1.4.1

Multiply $- 4$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8 \cdot 1}}{2 \cdot - 2}$

Step 2.4.4.7.1.4.2

Multiply $8$ by $1$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{2 \cdot - 2}$

Step 2.4.4.7.2

Multiply $2$ by $- 2$ .

$u = \frac{x \pm \sqrt{x^{2} + 8}}{- 4}$

Step 2.4.4.7.3

Move the negative in front of the fraction.

$u = - \frac{x \pm \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.7.4

Change the $\pm$ to $-$ .

$u = - \frac{x - \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.8

The final answer is the combination of both solutions.

$u = - \frac{x + \sqrt{x^{2} + 8}}{4}$

$u = - \frac{x - \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.9

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

$\sin (y) = - \frac{x + \sqrt{x^{2} + 8}}{4}, - \frac{x - \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.10

Set up each of the solutions to solve for $y$ .

$\sin (y) = - \frac{x + \sqrt{x^{2} + 8}}{4}$

$\sin (y) = - \frac{x - \sqrt{x^{2} + 8}}{4}$

Step 2.4.4.11

Solve for $y$ in $\sin (y) = - \frac{x + \sqrt{x^{2} + 8}}{4}$ .

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Step 2.4.4.11.1

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

$y = \arcsin (- \frac{x + \sqrt{x^{2} + 8}}{4})$

Step 2.4.4.11.2

Simplify the right side.

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Step 2.4.4.11.2.1

Simplify $\arcsin (- \frac{x + \sqrt{x^{2} + 8}}{4})$ .

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Step 2.4.4.11.2.1.1

Split the fraction $\frac{x + \sqrt{x^{2} + 8}}{4}$ into two fractions.

$y = \arcsin (- (\frac{x}{4} + \frac{\sqrt{x^{2} + 8}}{4}))$

Step 2.4.4.11.2.1.2

Apply the distributive property.

$y = \arcsin (- \frac{x}{4} - \frac{\sqrt{x^{2} + 8}}{4})$

Step 2.4.4.11.3

Simplify the right side.

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Step 2.4.4.11.3.1

Combine the numerators over the common denominator.

$y = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$

Step 2.4.4.12

Solve for $y$ in $\sin (y) = - \frac{x - \sqrt{x^{2} + 8}}{4}$ .

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Step 2.4.4.12.1

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

$y = \arcsin (- \frac{x - \sqrt{x^{2} + 8}}{4})$

Step 2.4.4.12.2

Simplify the right side.

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Step 2.4.4.12.2.1

Simplify $\arcsin (- \frac{x - \sqrt{x^{2} + 8}}{4})$ .

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Step 2.4.4.12.2.1.1

Split the fraction $\frac{x - \sqrt{x^{2} + 8}}{4}$ into two fractions.

$y = \arcsin (- (\frac{x}{4} + \frac{- \sqrt{x^{2} + 8}}{4}))$

Step 2.4.4.12.2.1.2

Move the negative in front of the fraction.

$y = \arcsin (- (\frac{x}{4} - \frac{\sqrt{x^{2} + 8}}{4}))$

Step 2.4.4.12.2.1.3

Apply the distributive property.

$y = \arcsin (- \frac{x}{4} - - \frac{\sqrt{x^{2} + 8}}{4})$

Step 2.4.4.12.2.1.4

Multiply $- - \frac{\sqrt{x^{2} + 8}}{4}$ .

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Step 2.4.4.12.2.1.4.1

Multiply $- 1$ by $- 1$ .

$y = \arcsin (- \frac{x}{4} + 1 \frac{\sqrt{x^{2} + 8}}{4})$

Step 2.4.4.12.2.1.4.2

Multiply $\frac{\sqrt{x^{2} + 8}}{4}$ by $1$ .

$y = \arcsin (- \frac{x}{4} + \frac{\sqrt{x^{2} + 8}}{4})$

Step 2.4.4.12.3

Simplify the right side.

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Step 2.4.4.12.3.1

Combine the numerators over the common denominator.

$y = \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

Step 2.4.4.13

List all of the solutions.

$y = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$

$y = \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

Step 3

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

$f^{- 1} (x) = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4}), \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$

Step 4

Verify if $f^{- 1} (x) = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4}), \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ is the inverse of $f (x) = \frac{\cos (2 x)}{\sin (x)}$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{\cos (2 x)}{\sin (x)}$ and $f^{- 1} (x) = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4}), \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ and compare them.

Step 4.2

Find the range of $f (x) = \frac{\cos (2 x)}{\sin (x)}$ .

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Step 4.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - 1] \cup [1, \infty)$

Step 4.3

Find the domain of the inverse.

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Step 4.3.1

Find the domain of $\arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$ .

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Step 4.3.1.1

Set the radicand in $\sqrt{x^{2} + 8}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 8 \geq 0$

Step 4.3.1.2

Solve for $x$ .

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Step 4.3.1.2.1

Subtract $8$ from both sides of the inequality.

$x^{2} \geq - 8$

Step 4.3.1.2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.3.1.3

Set the argument in $\arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{- x - \sqrt{x^{2} + 8}}{4} \geq - 1$

Step 4.3.1.4

Solve for $x$ .

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Step 4.3.1.4.1

Multiply both sides by $4$ .

$\frac{- x - \sqrt{x^{2} + 8}}{4} \cdot 4 \geq - 1 \cdot 4$

Step 4.3.1.4.2

Simplify.

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Step 4.3.1.4.2.1

Simplify the left side.

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Step 4.3.1.4.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.1.4.2.1.1.1

Cancel the common factor.

$\frac{- x - \sqrt{x^{2} + 8}}{4} \cdot 4 \geq - 1 \cdot 4$

Step 4.3.1.4.2.1.1.2

Rewrite the expression.

$- x - \sqrt{x^{2} + 8} \geq - 1 \cdot 4$

Step 4.3.1.4.2.2

Simplify the right side.

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Step 4.3.1.4.2.2.1

Multiply $- 1$ by $4$ .

$- x - \sqrt{x^{2} + 8} \geq - 4$

Step 4.3.1.4.3

Solve for $x$ .

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Step 4.3.1.4.3.1

Add $x$ to both sides of the inequality.

$- \sqrt{x^{2} + 8} \geq - 4 + x$

Step 4.3.1.4.3.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(- \sqrt{x^{2} + 8})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3

Simplify each side of the inequality.

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Step 4.3.1.4.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

${(- {(x^{2} + 8)}^{\frac{1}{2}})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2

Simplify the left side.

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Step 4.3.1.4.3.3.2.1

Simplify ${(- {(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.1.4.3.3.2.1.1

Apply the product rule to $- {(x^{2} + 8)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.3

Multiply ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.4

Multiply the exponents in ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.1.4.3.3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.3.1.4.3.3.2.1.4.2.1

Cancel the common factor.

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.4.2.2

Rewrite the expression.

${(x^{2} + 8)}^{1} \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.2.1.5

Simplify.

$x^{2} + 8 \geq {(- 4 + x)}^{2}$

Step 4.3.1.4.3.3.3

Simplify the right side.

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Step 4.3.1.4.3.3.3.1

Simplify ${(- 4 + x)}^{2}$ .

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Step 4.3.1.4.3.3.3.1.1

Rewrite ${(- 4 + x)}^{2}$ as $(- 4 + x) (- 4 + x)$ .

$x^{2} + 8 \geq (- 4 + x) (- 4 + x)$

Step 4.3.1.4.3.3.3.1.2

Expand $(- 4 + x) (- 4 + x)$ using the FOIL Method.

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Step 4.3.1.4.3.3.3.1.2.1

Apply the distributive property.

$x^{2} + 8 \geq - 4 (- 4 + x) + x (- 4 + x)$

Step 4.3.1.4.3.3.3.1.2.2

Apply the distributive property.

$x^{2} + 8 \geq - 4 \cdot - 4 - 4 x + x (- 4 + x)$

Step 4.3.1.4.3.3.3.1.2.3

Apply the distributive property.

$x^{2} + 8 \geq - 4 \cdot - 4 - 4 x + x \cdot - 4 + x \cdot x$

Step 4.3.1.4.3.3.3.1.3

Simplify and combine like terms.

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Step 4.3.1.4.3.3.3.1.3.1

Simplify each term.

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Step 4.3.1.4.3.3.3.1.3.1.1

Multiply $- 4$ by $- 4$ .

$x^{2} + 8 \geq 16 - 4 x + x \cdot - 4 + x \cdot x$

Step 4.3.1.4.3.3.3.1.3.1.2

Move $- 4$ to the left of $x$ .

$x^{2} + 8 \geq 16 - 4 x - 4 \cdot x + x \cdot x$

Step 4.3.1.4.3.3.3.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} + 8 \geq 16 - 4 x - 4 x + x^{2}$

Step 4.3.1.4.3.3.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} + 8 \geq 16 - 8 x + x^{2}$

Step 4.3.1.4.3.4

Solve for $x$ .

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Step 4.3.1.4.3.4.1

Rewrite so $x$ is on the left side of the inequality.

$16 - 8 x + x^{2} \leq x^{2} + 8$

Step 4.3.1.4.3.4.2

Move all terms containing $x$ to the left side of the inequality.

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Step 4.3.1.4.3.4.2.1

Subtract $x^{2}$ from both sides of the inequality.

$16 - 8 x + x^{2} - x^{2} \leq 8$

Step 4.3.1.4.3.4.2.2

Combine the opposite terms in $16 - 8 x + x^{2} - x^{2}$ .

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Step 4.3.1.4.3.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$16 - 8 x + 0 \leq 8$

Step 4.3.1.4.3.4.2.2.2

Add $16 - 8 x$ and $0$ .

$16 - 8 x \leq 8$

Step 4.3.1.4.3.4.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.3.1.4.3.4.3.1

Subtract $16$ from both sides of the inequality.

$- 8 x \leq 8 - 16$

Step 4.3.1.4.3.4.3.2

Subtract $16$ from $8$ .

$- 8 x \leq - 8$

Step 4.3.1.4.3.4.4

Divide each term in $- 8 x \leq - 8$ by $- 8$ and simplify.

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Step 4.3.1.4.3.4.4.1

Divide each term in $- 8 x \leq - 8$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \geq \frac{- 8}{- 8}$

Step 4.3.1.4.3.4.4.2

Simplify the left side.

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Step 4.3.1.4.3.4.4.2.1

Cancel the common factor of $- 8$ .

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Step 4.3.1.4.3.4.4.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \geq \frac{- 8}{- 8}$

Step 4.3.1.4.3.4.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 8}{- 8}$

Step 4.3.1.4.3.4.4.3

Simplify the right side.

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Step 4.3.1.4.3.4.4.3.1

Divide $- 8$ by $- 8$ .

$x \geq 1$

Step 4.3.1.4.4

The solution consists of all of the true intervals.

$x \leq 1$

Step 4.3.1.5

Set the argument in $\arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4})$ less than or equal to $1$ to find where the expression is defined.

$\frac{- x - \sqrt{x^{2} + 8}}{4} \leq 1$

Step 4.3.1.6

Solve for $x$ .

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Step 4.3.1.6.1

Multiply both sides by $4$ .

$\frac{- x - \sqrt{x^{2} + 8}}{4} \cdot 4 \leq 1 \cdot 4$

Step 4.3.1.6.2

Simplify.

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Step 4.3.1.6.2.1

Simplify the left side.

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Step 4.3.1.6.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.1.6.2.1.1.1

Cancel the common factor.

$\frac{- x - \sqrt{x^{2} + 8}}{4} \cdot 4 \leq 1 \cdot 4$

Step 4.3.1.6.2.1.1.2

Rewrite the expression.

$- x - \sqrt{x^{2} + 8} \leq 1 \cdot 4$

Step 4.3.1.6.2.2

Simplify the right side.

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Step 4.3.1.6.2.2.1

Multiply $4$ by $1$ .

$- x - \sqrt{x^{2} + 8} \leq 4$

Step 4.3.1.6.3

Solve for $x$ .

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Step 4.3.1.6.3.1

Add $x$ to both sides of the inequality.

$- \sqrt{x^{2} + 8} \leq 4 + x$

Step 4.3.1.6.3.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(- \sqrt{x^{2} + 8})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3

Simplify each side of the inequality.

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Step 4.3.1.6.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

${(- {(x^{2} + 8)}^{\frac{1}{2}})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2

Simplify the left side.

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Step 4.3.1.6.3.3.2.1

Simplify ${(- {(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.1.6.3.3.2.1.1

Apply the product rule to $- {(x^{2} + 8)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.3

Multiply ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.4

Multiply the exponents in ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.1.6.3.3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.3.1.6.3.3.2.1.4.2.1

Cancel the common factor.

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.4.2.2

Rewrite the expression.

${(x^{2} + 8)}^{1} \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.2.1.5

Simplify.

$x^{2} + 8 \leq {(4 + x)}^{2}$

Step 4.3.1.6.3.3.3

Simplify the right side.

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Step 4.3.1.6.3.3.3.1

Simplify ${(4 + x)}^{2}$ .

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Step 4.3.1.6.3.3.3.1.1

Rewrite ${(4 + x)}^{2}$ as $(4 + x) (4 + x)$ .

$x^{2} + 8 \leq (4 + x) (4 + x)$

Step 4.3.1.6.3.3.3.1.2

Expand $(4 + x) (4 + x)$ using the FOIL Method.

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Step 4.3.1.6.3.3.3.1.2.1

Apply the distributive property.

$x^{2} + 8 \leq 4 (4 + x) + x (4 + x)$

Step 4.3.1.6.3.3.3.1.2.2

Apply the distributive property.

$x^{2} + 8 \leq 4 \cdot 4 + 4 x + x (4 + x)$

Step 4.3.1.6.3.3.3.1.2.3

Apply the distributive property.

$x^{2} + 8 \leq 4 \cdot 4 + 4 x + x \cdot 4 + x \cdot x$

Step 4.3.1.6.3.3.3.1.3

Simplify and combine like terms.

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Step 4.3.1.6.3.3.3.1.3.1

Simplify each term.

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Step 4.3.1.6.3.3.3.1.3.1.1

Multiply $4$ by $4$ .

$x^{2} + 8 \leq 16 + 4 x + x \cdot 4 + x \cdot x$

Step 4.3.1.6.3.3.3.1.3.1.2

Move $4$ to the left of $x$ .

$x^{2} + 8 \leq 16 + 4 x + 4 \cdot x + x \cdot x$

Step 4.3.1.6.3.3.3.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} + 8 \leq 16 + 4 x + 4 x + x^{2}$

Step 4.3.1.6.3.3.3.1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 \leq 16 + 8 x + x^{2}$

Step 4.3.1.6.3.4

Solve for $x$ .

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Step 4.3.1.6.3.4.1

Rewrite so $x$ is on the left side of the inequality.

$16 + 8 x + x^{2} \geq x^{2} + 8$

Step 4.3.1.6.3.4.2

Move all terms containing $x$ to the left side of the inequality.

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Step 4.3.1.6.3.4.2.1

Subtract $x^{2}$ from both sides of the inequality.

$16 + 8 x + x^{2} - x^{2} \geq 8$

Step 4.3.1.6.3.4.2.2

Combine the opposite terms in $16 + 8 x + x^{2} - x^{2}$ .

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Step 4.3.1.6.3.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$16 + 8 x + 0 \geq 8$

Step 4.3.1.6.3.4.2.2.2

Add $16 + 8 x$ and $0$ .

$16 + 8 x \geq 8$

Step 4.3.1.6.3.4.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.3.1.6.3.4.3.1

Subtract $16$ from both sides of the inequality.

$8 x \geq 8 - 16$

Step 4.3.1.6.3.4.3.2

Subtract $16$ from $8$ .

$8 x \geq - 8$

Step 4.3.1.6.3.4.4

Divide each term in $8 x \geq - 8$ by $8$ and simplify.

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Step 4.3.1.6.3.4.4.1

Divide each term in $8 x \geq - 8$ by $8$ .

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Step 4.3.1.6.3.4.4.2

Simplify the left side.

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Step 4.3.1.6.3.4.4.2.1

Cancel the common factor of $8$ .

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Step 4.3.1.6.3.4.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Step 4.3.1.6.3.4.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 8}{8}$

Step 4.3.1.6.3.4.4.3

Simplify the right side.

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Step 4.3.1.6.3.4.4.3.1

Divide $- 8$ by $8$ .

$x \geq - 1$

Step 4.3.1.6.4

Use each root to create test intervals.

$x < - 1$

$x > - 1$

Step 4.3.1.6.5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.3.1.6.5.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

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Step 4.3.1.6.5.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 4.3.1.6.5.1.2

Replace $x$ with $- 4$ in the original inequality.

$\frac{- (- 4) - \sqrt{{(- 4)}^{2} + 8}}{4} \leq 1$

Step 4.3.1.6.5.1.3

The left side $- 0.22474487$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 4.3.1.6.5.2

Test a value on the interval $x > - 1$ to see if it makes the inequality true.

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Step 4.3.1.6.5.2.1

Choose a value on the interval $x > - 1$ and see if this value makes the original inequality true.

$x = 0$

Step 4.3.1.6.5.2.2

Replace $x$ with $0$ in the original inequality.

$\frac{- (0) - \sqrt{{(0)}^{2} + 8}}{4} \leq 1$

Step 4.3.1.6.5.2.3

The left side $- 0.70710678$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 4.3.1.6.5.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ True

$x > - 1$ True

$x < - 1$ True

$x > - 1$ True

Step 4.3.1.6.6

The solution consists of all of the true intervals.

$x \leq - 1$ or $x \geq - 1$

Step 4.3.1.6.7

Combine the intervals.

All real numbers

Step 4.3.1.7

The domain is all values of $x$ that make the expression defined.

$(- \infty, 1]$

Step 4.3.2

Find the domain of $\arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ .

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Step 4.3.2.1

Set the radicand in $\sqrt{x^{2} + 8}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 8 \geq 0$

Step 4.3.2.2

Solve for $x$ .

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Step 4.3.2.2.1

Subtract $8$ from both sides of the inequality.

$x^{2} \geq - 8$

Step 4.3.2.2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.3.2.3

Set the argument in $\arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{- x + \sqrt{x^{2} + 8}}{4} \geq - 1$

Step 4.3.2.4

Solve for $x$ .

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Step 4.3.2.4.1

Multiply both sides by $4$ .

$\frac{- x + \sqrt{x^{2} + 8}}{4} \cdot 4 \geq - 1 \cdot 4$

Step 4.3.2.4.2

Simplify.

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Step 4.3.2.4.2.1

Simplify the left side.

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Step 4.3.2.4.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.2.4.2.1.1.1

Cancel the common factor.

$\frac{- x + \sqrt{x^{2} + 8}}{4} \cdot 4 \geq - 1 \cdot 4$

Step 4.3.2.4.2.1.1.2

Rewrite the expression.

$- x + \sqrt{x^{2} + 8} \geq - 1 \cdot 4$

Step 4.3.2.4.2.2

Simplify the right side.

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Step 4.3.2.4.2.2.1

Multiply $- 1$ by $4$ .

$- x + \sqrt{x^{2} + 8} \geq - 4$

Step 4.3.2.4.3

Solve for $x$ .

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Step 4.3.2.4.3.1

Add $x$ to both sides of the inequality.

$\sqrt{x^{2} + 8} \geq - 4 + x$

Step 4.3.2.4.3.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x^{2} + 8}}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3

Simplify each side of the inequality.

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Step 4.3.2.4.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

${({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3.2

Simplify the left side.

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Step 4.3.2.4.3.3.2.1

Simplify ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.4.3.3.2.1.1

Multiply the exponents in ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.4.3.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.4.3.3.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 8)}^{1} \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3.2.1.2

Simplify.

$x^{2} + 8 \geq {(- 4 + x)}^{2}$

Step 4.3.2.4.3.3.3

Simplify the right side.

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Step 4.3.2.4.3.3.3.1

Simplify ${(- 4 + x)}^{2}$ .

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Step 4.3.2.4.3.3.3.1.1

Rewrite ${(- 4 + x)}^{2}$ as $(- 4 + x) (- 4 + x)$ .

$x^{2} + 8 \geq (- 4 + x) (- 4 + x)$

Step 4.3.2.4.3.3.3.1.2

Expand $(- 4 + x) (- 4 + x)$ using the FOIL Method.

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Step 4.3.2.4.3.3.3.1.2.1

Apply the distributive property.

$x^{2} + 8 \geq - 4 (- 4 + x) + x (- 4 + x)$

Step 4.3.2.4.3.3.3.1.2.2

Apply the distributive property.

$x^{2} + 8 \geq - 4 \cdot - 4 - 4 x + x (- 4 + x)$

Step 4.3.2.4.3.3.3.1.2.3

Apply the distributive property.

$x^{2} + 8 \geq - 4 \cdot - 4 - 4 x + x \cdot - 4 + x \cdot x$

Step 4.3.2.4.3.3.3.1.3

Simplify and combine like terms.

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Step 4.3.2.4.3.3.3.1.3.1

Simplify each term.

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Step 4.3.2.4.3.3.3.1.3.1.1

Multiply $- 4$ by $- 4$ .

$x^{2} + 8 \geq 16 - 4 x + x \cdot - 4 + x \cdot x$

Step 4.3.2.4.3.3.3.1.3.1.2

Move $- 4$ to the left of $x$ .

$x^{2} + 8 \geq 16 - 4 x - 4 \cdot x + x \cdot x$

Step 4.3.2.4.3.3.3.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} + 8 \geq 16 - 4 x - 4 x + x^{2}$

Step 4.3.2.4.3.3.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} + 8 \geq 16 - 8 x + x^{2}$

Step 4.3.2.4.3.4

Solve for $x$ .

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Step 4.3.2.4.3.4.1

Rewrite so $x$ is on the left side of the inequality.

$16 - 8 x + x^{2} \leq x^{2} + 8$

Step 4.3.2.4.3.4.2

Move all terms containing $x$ to the left side of the inequality.

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Step 4.3.2.4.3.4.2.1

Subtract $x^{2}$ from both sides of the inequality.

$16 - 8 x + x^{2} - x^{2} \leq 8$

Step 4.3.2.4.3.4.2.2

Combine the opposite terms in $16 - 8 x + x^{2} - x^{2}$ .

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Step 4.3.2.4.3.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$16 - 8 x + 0 \leq 8$

Step 4.3.2.4.3.4.2.2.2

Add $16 - 8 x$ and $0$ .

$16 - 8 x \leq 8$

Step 4.3.2.4.3.4.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.3.2.4.3.4.3.1

Subtract $16$ from both sides of the inequality.

$- 8 x \leq 8 - 16$

Step 4.3.2.4.3.4.3.2

Subtract $16$ from $8$ .

$- 8 x \leq - 8$

Step 4.3.2.4.3.4.4

Divide each term in $- 8 x \leq - 8$ by $- 8$ and simplify.

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Step 4.3.2.4.3.4.4.1

Divide each term in $- 8 x \leq - 8$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \geq \frac{- 8}{- 8}$

Step 4.3.2.4.3.4.4.2

Simplify the left side.

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Step 4.3.2.4.3.4.4.2.1

Cancel the common factor of $- 8$ .

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Step 4.3.2.4.3.4.4.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \geq \frac{- 8}{- 8}$

Step 4.3.2.4.3.4.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 8}{- 8}$

Step 4.3.2.4.3.4.4.3

Simplify the right side.

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Step 4.3.2.4.3.4.4.3.1

Divide $- 8$ by $- 8$ .

$x \geq 1$

Step 4.3.2.4.4

Use each root to create test intervals.

$x < 1$

$x > 1$

Step 4.3.2.4.5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.3.2.4.5.1

Test a value on the interval $x < 1$ to see if it makes the inequality true.

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Step 4.3.2.4.5.1.1

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 4.3.2.4.5.1.2

Replace $x$ with $0$ in the original inequality.

$\frac{- (0) + \sqrt{{(0)}^{2} + 8}}{4} \geq - 1$

Step 4.3.2.4.5.1.3

The left side $0.70710678$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 4.3.2.4.5.2

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 4.3.2.4.5.2.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 4.3.2.4.5.2.2

Replace $x$ with $4$ in the original inequality.

$\frac{- (4) + \sqrt{{(4)}^{2} + 8}}{4} \geq - 1$

Step 4.3.2.4.5.2.3

The left side $0.22474487$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 4.3.2.4.5.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ True

$x > 1$ True

$x < 1$ True

$x > 1$ True

Step 4.3.2.4.6

The solution consists of all of the true intervals.

$x \leq 1$ or $x \geq 1$

Step 4.3.2.4.7

Combine the intervals.

All real numbers

Step 4.3.2.5

Set the argument in $\arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ less than or equal to $1$ to find where the expression is defined.

$\frac{- x + \sqrt{x^{2} + 8}}{4} \leq 1$

Step 4.3.2.6

Solve for $x$ .

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Step 4.3.2.6.1

Multiply both sides by $4$ .

$\frac{- x + \sqrt{x^{2} + 8}}{4} \cdot 4 \leq 1 \cdot 4$

Step 4.3.2.6.2

Simplify.

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Step 4.3.2.6.2.1

Simplify the left side.

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Step 4.3.2.6.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.2.6.2.1.1.1

Cancel the common factor.

$\frac{- x + \sqrt{x^{2} + 8}}{4} \cdot 4 \leq 1 \cdot 4$

Step 4.3.2.6.2.1.1.2

Rewrite the expression.

$- x + \sqrt{x^{2} + 8} \leq 1 \cdot 4$

Step 4.3.2.6.2.2

Simplify the right side.

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Step 4.3.2.6.2.2.1

Multiply $4$ by $1$ .

$- x + \sqrt{x^{2} + 8} \leq 4$

Step 4.3.2.6.3

Solve for $x$ .

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Step 4.3.2.6.3.1

Add $x$ to both sides of the inequality.

$\sqrt{x^{2} + 8} \leq 4 + x$

Step 4.3.2.6.3.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x^{2} + 8}}^{2} \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3

Simplify each side of the inequality.

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Step 4.3.2.6.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

${({(x^{2} + 8)}^{\frac{1}{2}})}^{2} \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3.2

Simplify the left side.

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Step 4.3.2.6.3.3.2.1

Simplify ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.6.3.3.2.1.1

Multiply the exponents in ${({(x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.6.3.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.6.3.3.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 8)}^{\frac{1}{2} \cdot 2} \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 8)}^{1} \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3.2.1.2

Simplify.

$x^{2} + 8 \leq {(4 + x)}^{2}$

Step 4.3.2.6.3.3.3

Simplify the right side.

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Step 4.3.2.6.3.3.3.1

Simplify ${(4 + x)}^{2}$ .

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Step 4.3.2.6.3.3.3.1.1

Rewrite ${(4 + x)}^{2}$ as $(4 + x) (4 + x)$ .

$x^{2} + 8 \leq (4 + x) (4 + x)$

Step 4.3.2.6.3.3.3.1.2

Expand $(4 + x) (4 + x)$ using the FOIL Method.

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Step 4.3.2.6.3.3.3.1.2.1

Apply the distributive property.

$x^{2} + 8 \leq 4 (4 + x) + x (4 + x)$

Step 4.3.2.6.3.3.3.1.2.2

Apply the distributive property.

$x^{2} + 8 \leq 4 \cdot 4 + 4 x + x (4 + x)$

Step 4.3.2.6.3.3.3.1.2.3

Apply the distributive property.

$x^{2} + 8 \leq 4 \cdot 4 + 4 x + x \cdot 4 + x \cdot x$

Step 4.3.2.6.3.3.3.1.3

Simplify and combine like terms.

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Step 4.3.2.6.3.3.3.1.3.1

Simplify each term.

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Step 4.3.2.6.3.3.3.1.3.1.1

Multiply $4$ by $4$ .

$x^{2} + 8 \leq 16 + 4 x + x \cdot 4 + x \cdot x$

Step 4.3.2.6.3.3.3.1.3.1.2

Move $4$ to the left of $x$ .

$x^{2} + 8 \leq 16 + 4 x + 4 \cdot x + x \cdot x$

Step 4.3.2.6.3.3.3.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} + 8 \leq 16 + 4 x + 4 x + x^{2}$

Step 4.3.2.6.3.3.3.1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 \leq 16 + 8 x + x^{2}$

Step 4.3.2.6.3.4

Solve for $x$ .

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Step 4.3.2.6.3.4.1

Rewrite so $x$ is on the left side of the inequality.

$16 + 8 x + x^{2} \geq x^{2} + 8$

Step 4.3.2.6.3.4.2

Move all terms containing $x$ to the left side of the inequality.

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Step 4.3.2.6.3.4.2.1

Subtract $x^{2}$ from both sides of the inequality.

$16 + 8 x + x^{2} - x^{2} \geq 8$

Step 4.3.2.6.3.4.2.2

Combine the opposite terms in $16 + 8 x + x^{2} - x^{2}$ .

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Step 4.3.2.6.3.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$16 + 8 x + 0 \geq 8$

Step 4.3.2.6.3.4.2.2.2

Add $16 + 8 x$ and $0$ .

$16 + 8 x \geq 8$

Step 4.3.2.6.3.4.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.3.2.6.3.4.3.1

Subtract $16$ from both sides of the inequality.

$8 x \geq 8 - 16$

Step 4.3.2.6.3.4.3.2

Subtract $16$ from $8$ .

$8 x \geq - 8$

Step 4.3.2.6.3.4.4

Divide each term in $8 x \geq - 8$ by $8$ and simplify.

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Step 4.3.2.6.3.4.4.1

Divide each term in $8 x \geq - 8$ by $8$ .

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Step 4.3.2.6.3.4.4.2

Simplify the left side.

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Step 4.3.2.6.3.4.4.2.1

Cancel the common factor of $8$ .

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Step 4.3.2.6.3.4.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{- 8}{8}$

Step 4.3.2.6.3.4.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 8}{8}$

Step 4.3.2.6.3.4.4.3

Simplify the right side.

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Step 4.3.2.6.3.4.4.3.1

Divide $- 8$ by $8$ .

$x \geq - 1$

Step 4.3.2.7

The domain is all values of $x$ that make the expression defined.

$[- 1, \infty)$

Step 4.3.3

Find the union of $(- \infty, 1], [- 1, \infty)$ .

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Step 4.3.3.1

The union consists of all of the elements that are contained in each interval.

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4}), \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ is not equal to the range of $f (x) = \frac{\cos (2 x)}{\sin (x)}$ , then $f^{- 1} (x) = \arcsin (\frac{- x - \sqrt{x^{2} + 8}}{4}), \arcsin (\frac{- x + \sqrt{x^{2} + 8}}{4})$ is not an inverse of $f (x) = \frac{\cos (2 x)}{\sin (x)}$ .

There is no inverse

Step 5