Trigonometry Examples

${(4 - \cos (10 x))}^{- 11}$

Step 1

Interchange the variables.

$x = {(4 - \cos (10 y))}^{- 11}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as ${(4 - \cos (10 y))}^{- 11} = x$ .

${(4 - \cos (10 y))}^{- 11} = x$

Step 2.2

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

$\frac{1}{{(4 - \cos (10 y))}^{11}} = x$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

${(4 - \cos (10 y))}^{11}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

${(4 - \cos (10 y))}^{11}$

Step 2.4

Multiply each term in $\frac{1}{{(4 - \cos (10 y))}^{11}} = x$ by ${(4 - \cos (10 y))}^{11}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{{(4 - \cos (10 y))}^{11}} = x$ by ${(4 - \cos (10 y))}^{11}$ .

$\frac{1}{{(4 - \cos (10 y))}^{11}} {(4 - \cos (10 y))}^{11} = x {(4 - \cos (10 y))}^{11}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of ${(4 - \cos (10 y))}^{11}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{{(4 - \cos (10 y))}^{11}} {(4 - \cos (10 y))}^{11} = x {(4 - \cos (10 y))}^{11}$

Step 2.4.2.1.2

Rewrite the expression.

$1 = x {(4 - \cos (10 y))}^{11}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $x {(4 - \cos (10 y))}^{11} = 1$ .

$x {(4 - \cos (10 y))}^{11} = 1$

Step 2.5.2

Divide each term in $x {(4 - \cos (10 y))}^{11} = 1$ by $x$ and simplify.

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Step 2.5.2.1

Divide each term in $x {(4 - \cos (10 y))}^{11} = 1$ by $x$ .

$\frac{x {(4 - \cos (10 y))}^{11}}{x} = \frac{1}{x}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $x$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{x {(4 - \cos (10 y))}^{11}}{x} = \frac{1}{x}$

Step 2.5.2.2.1.2

Divide ${(4 - \cos (10 y))}^{11}$ by $1$ .

${(4 - \cos (10 y))}^{11} = \frac{1}{x}$

Step 2.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$4 - \cos (10 y) = \sqrt[11]{\frac{1}{x}}$

Step 2.5.4

Simplify $\sqrt[11]{\frac{1}{x}}$ .

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Step 2.5.4.1

Rewrite $\sqrt[11]{\frac{1}{x}}$ as $\frac{\sqrt[11]{1}}{\sqrt[11]{x}}$ .

$4 - \cos (10 y) = \frac{\sqrt[11]{1}}{\sqrt[11]{x}}$

Step 2.5.4.2

Any root of $1$ is $1$ .

$4 - \cos (10 y) = \frac{1}{\sqrt[11]{x}}$

Step 2.5.4.3

Multiply $\frac{1}{\sqrt[11]{x}}$ by $\frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{10}}$ .

$4 - \cos (10 y) = \frac{1}{\sqrt[11]{x}} \cdot \frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{10}}$

Step 2.5.4.4

Combine and simplify the denominator.

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Step 2.5.4.4.1

Multiply $\frac{1}{\sqrt[11]{x}}$ by $\frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{10}}$ .

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{\sqrt[11]{x} {\sqrt[11]{x}}^{10}}$

Step 2.5.4.4.2

Raise $\sqrt[11]{x}$ to the power of $1$ .

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{1} {\sqrt[11]{x}}^{10}}$

Step 2.5.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{1 + 10}}$

Step 2.5.4.4.4

Add $1$ and $10$ .

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{{\sqrt[11]{x}}^{11}}$

Step 2.5.4.4.5

Rewrite ${\sqrt[11]{x}}^{11}$ as $x$ .

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Step 2.5.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{x}$ as $x^{\frac{1}{11}}$ .

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{{(x^{\frac{1}{11}})}^{11}}$

Step 2.5.4.4.5.2

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

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{x^{\frac{1}{11} \cdot 11}}$

Step 2.5.4.4.5.3

Combine $\frac{1}{11}$ and $11$ .

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{x^{\frac{11}{11}}}$

Step 2.5.4.4.5.4

Cancel the common factor of $11$ .

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Step 2.5.4.4.5.4.1

Cancel the common factor.

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{x^{\frac{11}{11}}}$

Step 2.5.4.4.5.4.2

Rewrite the expression.

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{x^{1}}$

Step 2.5.4.4.5.5

Simplify.

$4 - \cos (10 y) = \frac{{\sqrt[11]{x}}^{10}}{x}$

Step 2.5.4.5

Rewrite ${\sqrt[11]{x}}^{10}$ as $\sqrt[11]{x^{10}}$ .

$4 - \cos (10 y) = \frac{\sqrt[11]{x^{10}}}{x}$

Step 2.5.5

Subtract $4$ from both sides of the equation.

$- \cos (10 y) = \frac{\sqrt[11]{x^{10}}}{x} - 4$

Step 2.5.6

Divide each term in $- \cos (10 y) = \frac{\sqrt[11]{x^{10}}}{x} - 4$ by $- 1$ and simplify.

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Step 2.5.6.1

Divide each term in $- \cos (10 y) = \frac{\sqrt[11]{x^{10}}}{x} - 4$ by $- 1$ .

$\frac{- \cos (10 y)}{- 1} = \frac{\frac{\sqrt[11]{x^{10}}}{x}}{- 1} + \frac{- 4}{- 1}$

Step 2.5.6.2

Simplify the left side.

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Step 2.5.6.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (10 y)}{1} = \frac{\frac{\sqrt[11]{x^{10}}}{x}}{- 1} + \frac{- 4}{- 1}$

Step 2.5.6.2.2

Divide $\cos (10 y)$ by $1$ .

$\cos (10 y) = \frac{\frac{\sqrt[11]{x^{10}}}{x}}{- 1} + \frac{- 4}{- 1}$

Step 2.5.6.3

Simplify the right side.

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Step 2.5.6.3.1

Simplify each term.

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Step 2.5.6.3.1.1

Move the negative one from the denominator of $\frac{\frac{\sqrt[11]{x^{10}}}{x}}{- 1}$ .

$\cos (10 y) = - 1 \cdot \frac{\sqrt[11]{x^{10}}}{x} + \frac{- 4}{- 1}$

Step 2.5.6.3.1.2

Rewrite $- 1 \cdot \frac{\sqrt[11]{x^{10}}}{x}$ as $- \frac{\sqrt[11]{x^{10}}}{x}$ .

$\cos (10 y) = - \frac{\sqrt[11]{x^{10}}}{x} + \frac{- 4}{- 1}$

Step 2.5.6.3.1.3

Divide $- 4$ by $- 1$ .

$\cos (10 y) = - \frac{\sqrt[11]{x^{10}}}{x} + 4$

Step 2.6

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

$10 y = \arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)$

Step 2.7

Divide each term in $10 y = \arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)$ by $10$ and simplify.

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Step 2.7.1

Divide each term in $10 y = \arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)$ by $10$ .

$\frac{10 y}{10} = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $10$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$

Step 2.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$ is the inverse of $f (x) = {(4 - \cos (10 x))}^{- 11}$ .

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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 $f^{- 1} ({(4 - \cos (10 x))}^{- 11})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{{({(4 - \cos (10 x))}^{- 11})}^{10}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Multiply the exponents in ${({(4 - \cos (10 x))}^{- 11})}^{10}$ .

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Step 4.2.3.1.1

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

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{{(4 - \cos (10 x))}^{- 11 \cdot 10}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.1.2

Multiply $- 11$ by $10$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{{(4 - \cos (10 x))}^{- 110}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.2

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

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{\frac{1}{{(4 - \cos (10 x))}^{110}}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.3

Rewrite $1$ as $1^{11}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{\frac{1^{11}}{{(4 - \cos (10 x))}^{110}}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.4

Rewrite ${(4 - \cos (10 x))}^{110}$ as ${({(4 - \cos (10 x))}^{10})}^{11}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{\frac{1^{11}}{{({(4 - \cos (10 x))}^{10})}^{11}}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.5

Rewrite $\frac{1^{11}}{{({(4 - \cos (10 x))}^{10})}^{11}}$ as ${(\frac{1}{{(4 - \cos (10 x))}^{10}})}^{11}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\sqrt[11]{{(\frac{1}{{(4 - \cos (10 x))}^{10}})}^{11}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.3.6

Pull terms out from under the radical, assuming real numbers.

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{\frac{1}{{(4 - \cos (10 x))}^{10}}}{{(4 - \cos (10 x))}^{- 11}} + 4)}{10}$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Simplify each term.

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Step 4.2.4.1.1

Move ${(4 - \cos (10 x))}^{- 11}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- (\frac{1}{{(4 - \cos (10 x))}^{10}} \cdot {(4 - \cos (10 x))}^{11}) + 4)}{10}$

Step 4.2.4.1.2

Combine $\frac{1}{{(4 - \cos (10 x))}^{10}}$ and ${(4 - \cos (10 x))}^{11}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{{(4 - \cos (10 x))}^{11}}{{(4 - \cos (10 x))}^{10}} + 4)}{10}$

Step 4.2.4.1.3

Cancel the common factor of ${(4 - \cos (10 x))}^{11}$ and ${(4 - \cos (10 x))}^{10}$ .

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Step 4.2.4.1.3.1

Factor ${(4 - \cos (10 x))}^{10}$ out of ${(4 - \cos (10 x))}^{11}$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{{(4 - \cos (10 x))}^{10} (4 - \cos (10 x))}{{(4 - \cos (10 x))}^{10}} + 4)}{10}$

Step 4.2.4.1.3.2

Cancel the common factors.

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Step 4.2.4.1.3.2.1

Multiply by $1$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{{(4 - \cos (10 x))}^{10} (4 - \cos (10 x))}{{(4 - \cos (10 x))}^{10} \cdot 1} + 4)}{10}$

Step 4.2.4.1.3.2.2

Cancel the common factor.

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{{(4 - \cos (10 x))}^{10} (4 - \cos (10 x))}{{(4 - \cos (10 x))}^{10} \cdot 1} + 4)}{10}$

Step 4.2.4.1.3.2.3

Rewrite the expression.

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- \frac{4 - \cos (10 x)}{1} + 4)}{10}$

Step 4.2.4.1.3.2.4

Divide $4 - \cos (10 x)$ by $1$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- (4 - \cos (10 x)) + 4)}{10}$

Step 4.2.4.1.4

Apply the distributive property.

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- 1 \cdot 4 + \cos (10 x) + 4)}{10}$

Step 4.2.4.1.5

Multiply $- 1$ by $4$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- 4 + \cos (10 x) + 4)}{10}$

Step 4.2.4.1.6

Multiply $- - \cos (10 x)$ .

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Step 4.2.4.1.6.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- 4 + 1 \cos (10 x) + 4)}{10}$

Step 4.2.4.1.6.2

Multiply $\cos (10 x)$ by $1$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (- 4 + \cos (10 x) + 4)}{10}$

Step 4.2.4.2

Combine the opposite terms in $- 4 + \cos (10 x) + 4$ .

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Step 4.2.4.2.1

Add $- 4$ and $4$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (0 + \cos (10 x))}{10}$

Step 4.2.4.2.2

Add $0$ and $\cos (10 x)$ .

$f^{- 1} ({(4 - \cos (10 x))}^{- 11}) = \frac{\arccos (\cos (10 x))}{10}$

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 $f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10})))}^{- 11}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Simplify the numerator.

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Step 4.3.3.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + \frac{4 x}{x})}{10})))}^{- 11}$

Step 4.3.3.1.2

Combine the numerators over the common denominator.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- \sqrt[11]{x^{10}} + 4 x}{x})}{10})))}^{- 11}$

Step 4.3.3.1.3

Simplify the numerator.

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Step 4.3.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{x^{10}}$ as $x^{\frac{10}{11}}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- x^{\frac{10}{11}} + 4 x}{x})}{10})))}^{- 11}$

Step 4.3.3.1.3.2

Factor $x^{\frac{10}{11}}$ out of $- x^{\frac{10}{11}} + 4 x$ .

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Step 4.3.3.1.3.2.1

Factor $x^{\frac{10}{11}}$ out of $- x^{\frac{10}{11}}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{x^{\frac{10}{11}} \cdot - 1 + 4 x}{x})}{10})))}^{- 11}$

Step 4.3.3.1.3.2.2

Factor $x^{\frac{10}{11}}$ out of $4 x$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{x^{\frac{10}{11}} \cdot - 1 + x^{\frac{10}{11}} (4 x^{\frac{1}{11}})}{x})}{10})))}^{- 11}$

Step 4.3.3.1.3.2.3

Factor $x^{\frac{10}{11}}$ out of $x^{\frac{10}{11}} \cdot - 1 + x^{\frac{10}{11}} (4 x^{\frac{1}{11}})$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{x^{\frac{10}{11}} (- 1 + 4 x^{\frac{1}{11}})}{x})}{10})))}^{- 11}$

Step 4.3.3.1.4

Move $x^{\frac{10}{11}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x \cdot x^{- \frac{10}{11}}})}{10})))}^{- 11}$

Step 4.3.3.1.5

Multiply $x$ by $x^{- \frac{10}{11}}$ by adding the exponents.

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Step 4.3.3.1.5.1

Multiply $x$ by $x^{- \frac{10}{11}}$ .

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Step 4.3.3.1.5.1.1

Raise $x$ to the power of $1$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x \cdot x^{- \frac{10}{11}}})}{10})))}^{- 11}$

Step 4.3.3.1.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{1 - \frac{10}{11}}})}{10})))}^{- 11}$

Step 4.3.3.1.5.2

Write $1$ as a fraction with a common denominator.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{11}{11} - \frac{10}{11}}})}{10})))}^{- 11}$

Step 4.3.3.1.5.3

Combine the numerators over the common denominator.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{11 - 10}{11}}})}{10})))}^{- 11}$

Step 4.3.3.1.5.4

Subtract $10$ from $11$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})}{10})))}^{- 11}$

Step 4.3.3.2

Cancel the common factor of $10$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (10 (\frac{\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})}{10})))}^{- 11}$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \cos (\arccos (\frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})))}^{- 11}$

Step 4.3.3.3

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(4 - \frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{11}}}{x^{\frac{1}{11}}}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}} - \frac{- 1 + 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.5

Combine the numerators over the common denominator.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{4 x^{\frac{1}{11}} - (- 1 + 4 x^{\frac{1}{11}})}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Apply the distributive property.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{4 x^{\frac{1}{11}} + 1 - (4 x^{\frac{1}{11}})}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.6.2

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{4 x^{\frac{1}{11}} + 1 - (4 x^{\frac{1}{11}})}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.6.3

Multiply $4$ by $- 1$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{4 x^{\frac{1}{11}} + 1 - 4 x^{\frac{1}{11}}}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.6.4

Subtract $4 x^{\frac{1}{11}}$ from $4 x^{\frac{1}{11}}$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{0 + 1}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.6.5

Add $0$ and $1$ .

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(\frac{1}{x^{\frac{1}{11}}})}^{- 11}$

Step 4.3.7

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = {(x^{\frac{1}{11}})}^{11}$

Step 4.3.8

Multiply the exponents in ${(x^{\frac{1}{11}})}^{11}$ .

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Step 4.3.8.1

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

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = x^{\frac{1}{11} \cdot 11}$

Step 4.3.8.2

Cancel the common factor of $11$ .

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Step 4.3.8.2.1

Cancel the common factor.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = x^{\frac{1}{11} \cdot 11}$

Step 4.3.8.2.2

Rewrite the expression.

$f (\frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$ is the inverse of $f (x) = {(4 - \cos (10 x))}^{- 11}$ .

$f^{- 1} (x) = \frac{\arccos (- \frac{\sqrt[11]{x^{10}}}{x} + 4)}{10}$