Trigonometry Examples

$\sec (\arctan (\frac{x}{3}))$

Step 1

Interchange the variables.

$x = \sec (\arctan (\frac{y}{3}))$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sec (\arctan (\frac{y}{3})) = x$ .

$\sec (\arctan (\frac{y}{3})) = x$

Step 2.2

Take the inverse secant of both sides of the equation to extract $\arctan (\frac{y}{3})$ from inside the secant.

$\arctan (\frac{y}{3}) = arcsec (x)$

Step 2.3

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

$\frac{y}{3} = \tan (arcsec (x))$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify $\tan (arcsec (x))$ .

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Step 2.4.1.1

Write the expression using exponents.

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Step 2.4.1.1.1

Draw a triangle in the plane with vertices $(1, \sqrt{x^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{x^{2} - 1^{2}})$ . Therefore, $\tan (arcsec (x))$ is $\sqrt{x^{2} - 1}$ .

$\frac{y}{3} = \sqrt{x^{2} - 1}$

Step 2.4.1.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{y}{3} = \sqrt{x^{2} - 1^{2}}$

Step 2.4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{y}{3} = \sqrt{(x + 1) (x - 1)}$

Step 2.5

Multiply both sides of the equation by $3$ .

$3 \frac{y}{3} = 3 \sqrt{(x + 1) (x - 1)}$

Step 2.6

Simplify the left side.

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Step 2.6.1

Cancel the common factor of $3$ .

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Step 2.6.1.1

Cancel the common factor.

$3 \frac{y}{3} = 3 \sqrt{(x + 1) (x - 1)}$

Step 2.6.1.2

Rewrite the expression.

$y = 3 \sqrt{(x + 1) (x - 1)}$

Step 3

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

$f^{- 1} (x) = 3 \sqrt{(x + 1) (x - 1)}$

Step 4

Verify if $f^{- 1} (x) = 3 \sqrt{(x + 1) (x - 1)}$ is the inverse of $f (x) = \sec (\arctan (\frac{x}{3}))$ .

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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 $f^{- 1} (\sec (\arctan (\frac{x}{3})))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{((\sec (\arctan (\frac{x}{3}))) + 1) ((\sec (\arctan (\frac{x}{3}))) - 1)}$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Draw a triangle in the plane with vertices $(1, \frac{x}{3})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{3})$ . Therefore, $\sec (\arctan (\frac{x}{3}))$ is $\sqrt{1 + {(\frac{x}{3})}^{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{1 + {(\frac{x}{3})}^{2}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.3.2

Apply the product rule to $\frac{x}{3}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{1 + \frac{x^{2}}{3^{2}}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.3.3

Raise $3$ to the power of $2$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{1 + \frac{x^{2}}{9}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.3.4

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{\frac{9}{9} + \frac{x^{2}}{9}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{\frac{9 + x^{2}}{9}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.5

Rewrite $\frac{9 + x^{2}}{9}$ as ${(\frac{1}{3})}^{2} (9 + x^{2})$ .

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Step 4.2.5.1

Factor the perfect power $1^{2}$ out of $9 + x^{2}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{\frac{1^{2} (9 + x^{2})}{9}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.5.2

Factor the perfect power $3^{2}$ out of $9$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{\frac{1^{2} (9 + x^{2})}{3^{2} \cdot 1}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.5.3

Rearrange the fraction $\frac{1^{2} (9 + x^{2})}{3^{2} \cdot 1}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\sqrt{{(\frac{1}{3})}^{2} (9 + x^{2})} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.6

Pull terms out from under the radical.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\frac{1}{3} \cdot \sqrt{9 + x^{2}} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.7

Combine $\frac{1}{3}$ and $\sqrt{9 + x^{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\frac{\sqrt{9 + x^{2}}}{3} + 1) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.8

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{(\frac{\sqrt{9 + x^{2}}}{3} + \frac{3}{3}) (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.9

Combine the numerators over the common denominator.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sec (\arctan (\frac{x}{3})) - 1)}$

Step 4.2.10

Simplify the expression.

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Step 4.2.10.1

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{1 + {(\frac{x}{3})}^{2}} - 1)}$

Step 4.2.10.2

Apply the product rule to $\frac{x}{3}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{1 + \frac{x^{2}}{3^{2}}} - 1)}$

Step 4.2.10.3

Raise $3$ to the power of $2$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{1 + \frac{x^{2}}{9}} - 1)}$

Step 4.2.10.4

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{\frac{9}{9} + \frac{x^{2}}{9}} - 1)}$

Step 4.2.11

Combine the numerators over the common denominator.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{\frac{9 + x^{2}}{9}} - 1)}$

Step 4.2.12

Rewrite $\frac{9 + x^{2}}{9}$ as ${(\frac{1}{3})}^{2} (9 + x^{2})$ .

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Step 4.2.12.1

Factor the perfect power $1^{2}$ out of $9 + x^{2}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{\frac{1^{2} (9 + x^{2})}{9}} - 1)}$

Step 4.2.12.2

Factor the perfect power $3^{2}$ out of $9$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{\frac{1^{2} (9 + x^{2})}{3^{2} \cdot 1}} - 1)}$

Step 4.2.12.3

Rearrange the fraction $\frac{1^{2} (9 + x^{2})}{3^{2} \cdot 1}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\sqrt{{(\frac{1}{3})}^{2} (9 + x^{2})} - 1)}$

Step 4.2.13

Pull terms out from under the radical.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\frac{1}{3} \cdot \sqrt{9 + x^{2}} - 1)}$

Step 4.2.14

Combine $\frac{1}{3}$ and $\sqrt{9 + x^{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\frac{\sqrt{9 + x^{2}}}{3} - 1)}$

Step 4.2.15

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\frac{\sqrt{9 + x^{2}}}{3} - 1 \cdot \frac{3}{3})}$

Step 4.2.16

Simplify terms.

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Step 4.2.16.1

Combine $- 1$ and $\frac{3}{3}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot (\frac{\sqrt{9 + x^{2}}}{3} + \frac{- 1 \cdot 3}{3})}$

Step 4.2.16.2

Combine the numerators over the common denominator.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot \frac{\sqrt{9 + x^{2}} - 1 \cdot 3}{3}}$

Step 4.2.16.3

Multiply $- 1$ by $3$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} + 3}{3} \cdot \frac{\sqrt{9 + x^{2}} - 3}{3}}$

Step 4.2.16.4

Multiply $\frac{\sqrt{9 + x^{2}} + 3}{3}$ by $\frac{\sqrt{9 + x^{2}} - 3}{3}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{(\sqrt{9 + x^{2}} + 3) (\sqrt{9 + x^{2}} - 3)}{3 \cdot 3}}$

Step 4.2.16.5

Multiply $3$ by $3$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{(\sqrt{9 + x^{2}} + 3) (\sqrt{9 + x^{2}} - 3)}{9}}$

Step 4.2.17

Expand $(\sqrt{9 + x^{2}} + 3) (\sqrt{9 + x^{2}} - 3)$ using the FOIL Method.

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Step 4.2.17.1

Apply the distributive property.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} (\sqrt{9 + x^{2}} - 3) + 3 (\sqrt{9 + x^{2}} - 3)}{9}}$

Step 4.2.17.2

Apply the distributive property.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + \sqrt{9 + x^{2}} \cdot - 3 + 3 (\sqrt{9 + x^{2}} - 3)}{9}}$

Step 4.2.17.3

Apply the distributive property.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + \sqrt{9 + x^{2}} \cdot - 3 + 3 \sqrt{9 + x^{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18

Simplify terms.

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Step 4.2.18.1

Combine the opposite terms in $\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + \sqrt{9 + x^{2}} \cdot - 3 + 3 \sqrt{9 + x^{2}} + 3 \cdot - 3$ .

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Step 4.2.18.1.1

Reorder the factors in the terms $\sqrt{9 + x^{2}} \cdot - 3$ and $3 \sqrt{9 + x^{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} - 3 \sqrt{9 + x^{2}} + 3 \sqrt{9 + x^{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.1.2

Add $- 3 \sqrt{9 + x^{2}}$ and $3 \sqrt{9 + x^{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + 0 + 3 \cdot - 3}{9}}$

Step 4.2.18.1.3

Add $\sqrt{9 + x^{2}} \sqrt{9 + x^{2}}$ and $0$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.2

Simplify each term.

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Step 4.2.18.2.1

Multiply $\sqrt{9 + x^{2}} \sqrt{9 + x^{2}}$ .

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Step 4.2.18.2.1.1

Raise $\sqrt{9 + x^{2}}$ to the power of $1$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.1.2

Raise $\sqrt{9 + x^{2}}$ to the power of $1$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{\sqrt{9 + x^{2}} \sqrt{9 + x^{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.1.3

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{\sqrt{9 + x^{2}}}^{1 + 1} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.1.4

Add $1$ and $1$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{\sqrt{9 + x^{2}}}^{2} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2

Rewrite ${\sqrt{9 + x^{2}}}^{2}$ as $9 + x^{2}$ .

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Step 4.2.18.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + x^{2}}$ as ${(9 + x^{2})}^{\frac{1}{2}}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{({(9 + x^{2})}^{\frac{1}{2}})}^{2} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2.2

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{(9 + x^{2})}^{\frac{1}{2} \cdot 2} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{(9 + x^{2})}^{\frac{2}{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2.4

Cancel the common factor of $2$ .

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Step 4.2.18.2.2.4.1

Cancel the common factor.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{{(9 + x^{2})}^{\frac{2}{2}} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2.4.2

Rewrite the expression.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{(9 + x^{2}) + 3 \cdot - 3}{9}}$

Step 4.2.18.2.2.5

Simplify.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{9 + x^{2} + 3 \cdot - 3}{9}}$

Step 4.2.18.2.3

Multiply $3$ by $- 3$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{9 + x^{2} - 9}{9}}$

Step 4.2.18.3

Simplify by adding terms.

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Step 4.2.18.3.1

Combine the opposite terms in $9 + x^{2} - 9$ .

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Step 4.2.18.3.1.1

Subtract $9$ from $9$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{x^{2} + 0}{9}}$

Step 4.2.18.3.1.2

Add $x^{2}$ and $0$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{x^{2}}{9}}$

Step 4.2.18.3.2

Rewrite $9$ as $3^{2}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{\frac{x^{2}}{3^{2}}}$

Step 4.2.19

Rewrite $\frac{x^{2}}{3^{2}}$ as ${(\frac{x}{3})}^{2}$ .

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 \sqrt{{(\frac{x}{3})}^{2}}$

Step 4.2.20

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

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 (\frac{x}{3})$

Step 4.2.21

Cancel the common factor of $3$ .

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Step 4.2.21.1

Cancel the common factor.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = 3 (\frac{x}{3})$

Step 4.2.21.2

Rewrite the expression.

$f^{- 1} (\sec (\arctan (\frac{x}{3}))) = x$

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 $f (3 \sqrt{(x + 1) (x - 1)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sec (\arctan (\frac{3 \sqrt{(x + 1) (x - 1)}}{3}))$

Step 4.3.3

Simplify by cancelling exponent with radical.

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Step 4.3.3.1

Draw a triangle in the plane with vertices $(1, \frac{3 \sqrt{(x + 1) (x - 1)}}{3})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{3 \sqrt{(x + 1) (x - 1)}}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{3 \sqrt{(x + 1) (x - 1)}}{3})$ . Therefore, $\sec (\arctan (\frac{3 \sqrt{(x + 1) (x - 1)}}{3}))$ is $\sqrt{1 + {\sqrt{(x + 1) (x - 1)}}^{2}}$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + {\sqrt{(x + 1) (x - 1)}}^{2}}$

Step 4.3.3.2

Rewrite ${\sqrt{(x + 1) (x - 1)}}^{2}$ as $(x + 1) (x - 1)$ .

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Step 4.3.3.2.1

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

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + {({((x + 1) (x - 1))}^{\frac{1}{2}})}^{2}}$

Step 4.3.3.2.2

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

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + {((x + 1) (x - 1))}^{\frac{1}{2} \cdot 2}}$

Step 4.3.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + {((x + 1) (x - 1))}^{\frac{2}{2}}}$

Step 4.3.3.2.4

Cancel the common factor of $2$ .

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Step 4.3.3.2.4.1

Cancel the common factor.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + {((x + 1) (x - 1))}^{\frac{2}{2}}}$

Step 4.3.3.2.4.2

Rewrite the expression.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + (x + 1) (x - 1)}$

Step 4.3.3.2.5

Simplify.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + (x + 1) (x - 1)}$

Step 4.3.4

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 4.3.4.1

Apply the distributive property.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x (x - 1) + 1 (x - 1)}$

Step 4.3.4.2

Apply the distributive property.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x \cdot x + x \cdot - 1 + 1 (x - 1)}$

Step 4.3.4.3

Apply the distributive property.

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1}$

Step 4.3.5

Simplify and combine like terms.

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Step 4.3.5.1

Simplify each term.

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Step 4.3.5.1.1

Multiply $x$ by $x$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1}$

Step 4.3.5.1.2

Move $- 1$ to the left of $x$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1}$

Step 4.3.5.1.3

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

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} - x + 1 x + 1 \cdot - 1}$

Step 4.3.5.1.4

Multiply $x$ by $1$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} - x + x + 1 \cdot - 1}$

Step 4.3.5.1.5

Multiply $- 1$ by $1$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} - x + x - 1}$

Step 4.3.5.2

Add $- x$ and $x$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} + 0 - 1}$

Step 4.3.5.3

Add $x^{2}$ and $0$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{1 + x^{2} - 1}$

Step 4.3.6

Subtract $1$ from $1$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{x^{2} + 0}$

Step 4.3.7

Add $x^{2}$ and $0$ .

$f (3 \sqrt{(x + 1) (x - 1)}) = \sqrt{x^{2}}$

Step 4.3.8

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

$f (3 \sqrt{(x + 1) (x - 1)}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 3 \sqrt{(x + 1) (x - 1)}$ is the inverse of $f (x) = \sec (\arctan (\frac{x}{3}))$ .

$f^{- 1} (x) = 3 \sqrt{(x + 1) (x - 1)}$