Trigonometry Examples

$\cos (arccsc (u))$

Step 1

Interchange the variables.

$u = \cos (arccsc (y))$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\cos (arccsc (y)) = u$ .

$\cos (arccsc (y)) = u$

Step 2.2

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

$arccsc (y) = \arccos (u)$

Step 2.3

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

$y = \csc (\arccos (u))$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify $\csc (\arccos (u))$ .

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Step 2.4.1.1

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

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

Step 2.4.1.2

Simplify the denominator.

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Step 2.4.1.2.1

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

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

Step 2.4.1.2.2

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

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

Step 2.4.1.3

Multiply $\frac{1}{\sqrt{(1 + u) (1 - u)}}$ by $\frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$ .

$y = \frac{1}{\sqrt{(1 + u) (1 - u)}} \cdot \frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$

Step 2.4.1.4

Combine and simplify the denominator.

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Step 2.4.1.4.1

Multiply $\frac{1}{\sqrt{(1 + u) (1 - u)}}$ by $\frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$ .

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

Step 2.4.1.4.2

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1} \sqrt{(1 + u) (1 - u)}}$

Step 2.4.1.4.3

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1} {\sqrt{(1 + u) (1 - u)}}^{1}}$

Step 2.4.1.4.4

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

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1 + 1}}$

Step 2.4.1.4.5

Add $1$ and $1$ .

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{2}}$

Step 2.4.1.4.6

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

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Step 2.4.1.4.6.1

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

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{({((1 + u) (1 - u))}^{\frac{1}{2}})}^{2}}$

Step 2.4.1.4.6.2

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

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{((1 + u) (1 - u))}^{\frac{1}{2} \cdot 2}}$

Step 2.4.1.4.6.3

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

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{((1 + u) (1 - u))}^{\frac{2}{2}}}$

Step 2.4.1.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.1.4.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{((1 + u) (1 - u))}^{\frac{2}{2}}}$

Step 2.4.1.4.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{(1 + u) (1 - u)}}{{((1 + u) (1 - u))}^{1}}$

Step 2.4.1.4.6.5

Simplify.

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

Step 3

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

$f^{- 1} (u) = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$

Step 4

Verify if $f^{- 1} (u) = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$ is the inverse of $f (u) = \cos (arccsc (u))$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\cos (arccsc (u)))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(1 + \cos (arccsc (u))) (1 - (\cos (arccsc (u))))}}{(1 + \cos (arccsc (u))) (1 - (\cos (arccsc (u))))}$

Step 4.2.3

Remove parentheses.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(1 + \cos (arccsc (u))) (1 - (\cos (arccsc (u))))}}{(1 + \cos (arccsc (u))) (1 - (\cos (arccsc (u))))}$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(1 + \frac{\sqrt{u^{2} - 1}}{u}) (1 - \cos (arccsc (u)))}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.2

Simplify the numerator.

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Step 4.2.4.2.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(1 + \frac{\sqrt{u^{2} - 1^{2}}}{u}) (1 - \cos (arccsc (u)))}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.2.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(1 + \frac{\sqrt{(u + 1) (u - 1)}}{u}) (1 - \cos (arccsc (u)))}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{(\frac{u}{u} + \frac{\sqrt{(u + 1) (u - 1)}}{u}) (1 - \cos (arccsc (u)))}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \cos (arccsc (u)))}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.5

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{u^{2} - 1}}{u})}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.6

Simplify the numerator.

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Step 4.2.4.6.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{u^{2} - 1^{2}}}{u})}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.6.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{(u + 1) (u - 1)}}{u})}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.7

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (\frac{u}{u} - \frac{\sqrt{(u + 1) (u - 1)}}{u})}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.8

Combine the numerators over the common denominator.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot \frac{u - \sqrt{(u + 1) (u - 1)}}{u}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.9

Multiply $\frac{u + \sqrt{(u + 1) (u - 1)}}{u}$ by $\frac{u - \sqrt{(u + 1) (u - 1)}}{u}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})}{u \cdot u}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.10

Multiply $u$ by $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.11

Expand $(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})$ using the FOIL Method.

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Step 4.2.4.11.1

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u (u - \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.11.2

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.11.3

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.12

Combine the opposite terms in $u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})$ .

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Step 4.2.4.12.1

Reorder the factors in the terms $u (- \sqrt{(u + 1) (u - 1)})$ and $\sqrt{(u + 1) (u - 1)} u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u \cdot u - u \sqrt{(u + 1) (u - 1)} + u \sqrt{(u + 1) (u - 1)} + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.12.2

Add $- u \sqrt{(u + 1) (u - 1)}$ and $u \sqrt{(u + 1) (u - 1)}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u \cdot u + 0 + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.12.3

Add $u \cdot u$ and $0$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u \cdot u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13

Simplify each term.

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Step 4.2.4.13.1

Multiply $u$ by $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.2

Rewrite using the commutative property of multiplication.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - \sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.3

Multiply $- \sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)}$ .

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Step 4.2.4.13.3.1

Raise $\sqrt{(u + 1) (u - 1)}$ to the power of $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (\sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.3.2

Raise $\sqrt{(u + 1) (u - 1)}$ to the power of $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (\sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)})}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.3.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {\sqrt{(u + 1) (u - 1)}}^{1 + 1}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.3.4

Add $1$ and $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {\sqrt{(u + 1) (u - 1)}}^{2}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4

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

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Step 4.2.4.13.4.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {({((u + 1) (u - 1))}^{\frac{1}{2}})}^{2}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{1}{2} \cdot 2}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{2}{2}}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4.4

Cancel the common factor of $2$ .

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Step 4.2.4.13.4.4.1

Cancel the common factor.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{2}{2}}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4.4.2

Rewrite the expression.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - ((u + 1) (u - 1))}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.4.5

Simplify.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - ((u + 1) (u - 1))}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.5

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

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Step 4.2.4.13.5.1

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u (u - 1) + 1 (u - 1))}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.5.2

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u \cdot u + u \cdot - 1 + 1 (u - 1))}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.5.3

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u \cdot u + u \cdot - 1 + 1 u + 1 \cdot - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6

Simplify and combine like terms.

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Step 4.2.4.13.6.1

Simplify each term.

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Step 4.2.4.13.6.1.1

Multiply $u$ by $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} + u \cdot - 1 + 1 u + 1 \cdot - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.1.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} - 1 \cdot u + 1 u + 1 \cdot - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.1.3

Rewrite $- 1 u$ as $- u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} - u + 1 u + 1 \cdot - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.1.4

Multiply $u$ by $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} - u + u + 1 \cdot - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.1.5

Multiply $- 1$ by $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} - u + u - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.2

Add $- u$ and $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} + 0 - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.6.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - (u^{2} - 1)}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.7

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - u^{2} + 1}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.13.8

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{u^{2} - u^{2} + 1}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.14

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{0 + 1}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.15

Add $0$ and $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{1}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.16

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{\frac{1^{2}}{u^{2}}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.17

Rewrite $\frac{1^{2}}{u^{2}}$ as ${(\frac{1}{u})}^{2}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\sqrt{{(\frac{1}{u})}^{2}}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.4.18

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{(1 + \cos (arccsc (u))) (1 - \cos (arccsc (u)))}$

Step 4.2.5

Simplify the denominator.

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Step 4.2.5.1

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{(1 + \frac{\sqrt{u^{2} - 1}}{u}) (1 - \cos (arccsc (u)))}$

Step 4.2.5.2

Simplify the numerator.

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Step 4.2.5.2.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{(1 + \frac{\sqrt{u^{2} - 1^{2}}}{u}) (1 - \cos (arccsc (u)))}$

Step 4.2.5.2.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{(1 + \frac{\sqrt{(u + 1) (u - 1)}}{u}) (1 - \cos (arccsc (u)))}$

Step 4.2.5.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{(\frac{u}{u} + \frac{\sqrt{(u + 1) (u - 1)}}{u}) (1 - \cos (arccsc (u)))}$

Step 4.2.5.4

Combine the numerators over the common denominator.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \cos (arccsc (u)))}$

Step 4.2.5.5

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{u^{2} - 1}}{u})}$

Step 4.2.5.6

Simplify the numerator.

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Step 4.2.5.6.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{u^{2} - 1^{2}}}{u})}$

Step 4.2.5.6.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (1 - \frac{\sqrt{(u + 1) (u - 1)}}{u})}$

Step 4.2.5.7

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot (\frac{u}{u} - \frac{\sqrt{(u + 1) (u - 1)}}{u})}$

Step 4.2.5.8

Combine the numerators over the common denominator.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u + \sqrt{(u + 1) (u - 1)}}{u} \cdot \frac{u - \sqrt{(u + 1) (u - 1)}}{u}}$

Step 4.2.6

Multiply $\frac{u + \sqrt{(u + 1) (u - 1)}}{u}$ by $\frac{u - \sqrt{(u + 1) (u - 1)}}{u}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})}{u \cdot u}}$

Step 4.2.7

Simplify the denominator.

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Step 4.2.7.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})}{u^{1 + 1}}}$

Step 4.2.7.2

Add $1$ and $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8

Simplify the denominator.

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Step 4.2.8.1

Expand $(u + \sqrt{(u + 1) (u - 1)}) (u - \sqrt{(u + 1) (u - 1)})$ using the FOIL Method.

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Step 4.2.8.1.1

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u (u - \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.1.2

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} (u - \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.1.3

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.2

Combine the opposite terms in $u \cdot u + u (- \sqrt{(u + 1) (u - 1)}) + \sqrt{(u + 1) (u - 1)} u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})$ .

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Step 4.2.8.2.1

Reorder the factors in the terms $u (- \sqrt{(u + 1) (u - 1)})$ and $\sqrt{(u + 1) (u - 1)} u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u \cdot u - u \sqrt{(u + 1) (u - 1)} + u \sqrt{(u + 1) (u - 1)} + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.2.2

Add $- u \sqrt{(u + 1) (u - 1)}$ and $u \sqrt{(u + 1) (u - 1)}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u \cdot u + 0 + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.2.3

Add $u \cdot u$ and $0$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u \cdot u + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.3

Simplify each term.

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Step 4.2.8.3.1

Multiply $u$ by $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} + \sqrt{(u + 1) (u - 1)} (- \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.3.2

Rewrite using the commutative property of multiplication.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - \sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)}}{u^{2}}}$

Step 4.2.8.3.3

Multiply $- \sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)}$ .

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Step 4.2.8.3.3.1

Raise $\sqrt{(u + 1) (u - 1)}$ to the power of $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (\sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.3.3.2

Raise $\sqrt{(u + 1) (u - 1)}$ to the power of $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (\sqrt{(u + 1) (u - 1)} \sqrt{(u + 1) (u - 1)})}{u^{2}}}$

Step 4.2.8.3.3.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {\sqrt{(u + 1) (u - 1)}}^{1 + 1}}{u^{2}}}$

Step 4.2.8.3.3.4

Add $1$ and $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {\sqrt{(u + 1) (u - 1)}}^{2}}{u^{2}}}$

Step 4.2.8.3.4

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

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Step 4.2.8.3.4.1

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {({((u + 1) (u - 1))}^{\frac{1}{2}})}^{2}}{u^{2}}}$

Step 4.2.8.3.4.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{1}{2} \cdot 2}}{u^{2}}}$

Step 4.2.8.3.4.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{2}{2}}}{u^{2}}}$

Step 4.2.8.3.4.4

Cancel the common factor of $2$ .

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Step 4.2.8.3.4.4.1

Cancel the common factor.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - {((u + 1) (u - 1))}^{\frac{2}{2}}}{u^{2}}}$

Step 4.2.8.3.4.4.2

Rewrite the expression.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - ((u + 1) (u - 1))}{u^{2}}}$

Step 4.2.8.3.4.5

Simplify.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - ((u + 1) (u - 1))}{u^{2}}}$

Step 4.2.8.3.5

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

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Step 4.2.8.3.5.1

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u (u - 1) + 1 (u - 1))}{u^{2}}}$

Step 4.2.8.3.5.2

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u \cdot u + u \cdot - 1 + 1 (u - 1))}{u^{2}}}$

Step 4.2.8.3.5.3

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u \cdot u + u \cdot - 1 + 1 u + 1 \cdot - 1)}{u^{2}}}$

Step 4.2.8.3.6

Simplify and combine like terms.

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Step 4.2.8.3.6.1

Simplify each term.

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Step 4.2.8.3.6.1.1

Multiply $u$ by $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} + u \cdot - 1 + 1 u + 1 \cdot - 1)}{u^{2}}}$

Step 4.2.8.3.6.1.2

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} - 1 \cdot u + 1 u + 1 \cdot - 1)}{u^{2}}}$

Step 4.2.8.3.6.1.3

Rewrite $- 1 u$ as $- u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} - u + 1 u + 1 \cdot - 1)}{u^{2}}}$

Step 4.2.8.3.6.1.4

Multiply $u$ by $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} - u + u + 1 \cdot - 1)}{u^{2}}}$

Step 4.2.8.3.6.1.5

Multiply $- 1$ by $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} - u + u - 1)}{u^{2}}}$

Step 4.2.8.3.6.2

Add $- u$ and $u$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} + 0 - 1)}{u^{2}}}$

Step 4.2.8.3.6.3

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - (u^{2} - 1)}{u^{2}}}$

Step 4.2.8.3.7

Apply the distributive property.

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - u^{2} + 1}{u^{2}}}$

Step 4.2.8.3.8

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{u^{2} - u^{2} + 1}{u^{2}}}$

Step 4.2.8.4

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

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{0 + 1}{u^{2}}}$

Step 4.2.8.5

Add $0$ and $1$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{\frac{1}{u}}{\frac{1}{u^{2}}}$

Step 4.2.9

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\cos (arccsc (u))) = \frac{1}{u} \cdot u^{2}$

Step 4.2.10

Cancel the common factor of $u$ .

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Step 4.2.10.1

Factor $u$ out of $u^{2}$ .

$f^{- 1} (\cos (arccsc (u))) = \frac{1}{u} \cdot (u \cdot u)$

Step 4.2.10.2

Cancel the common factor.

$f^{- 1} (\cos (arccsc (u))) = \frac{1}{u} \cdot (u \cdot u)$

Step 4.2.10.3

Rewrite the expression.

$f^{- 1} (\cos (arccsc (u))) = u$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \cos (arccsc (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}))$

Step 4.3.3

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

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

Simplify.

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Step 4.3.7.1

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

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

Step 4.3.7.2

Combine the numerators over the common denominator.

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{\sqrt{(1 + u) (1 - u)} + (1 + u) (1 - u)}{(1 + u) (1 - u)} \cdot (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)} - 1)} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.3

Rewrite $\frac{\sqrt{(1 + u) (1 - u)} + (1 + u) (1 - u)}{(1 + u) (1 - u)}$ in a factored form.

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Step 4.3.7.3.1

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

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

Step 4.3.7.3.2

Rewrite $(1 + u) (1 - u)$ as ${({((1 + u) (1 - u))}^{\frac{1}{2}})}^{2}$ .

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

Step 4.3.7.3.3

Let $u = {((1 + u) (1 - u))}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${((1 + u) (1 - u))}^{\frac{1}{2}}$ .

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

Step 4.3.7.3.4

Factor $u$ out of $u + u^{2}$ .

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Step 4.3.7.3.4.1

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

Step 4.3.7.3.4.2

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

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

Step 4.3.7.3.4.3

Factor $u$ out of $u^{2}$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{u \cdot 1 + u \cdot u}{(1 + u) (1 - u)} \cdot (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)} - 1)} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.3.4.4

Factor $u$ out of $u \cdot 1 + u \cdot u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{u (1 + u)}{(1 + u) (1 - u)} \cdot (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)} - 1)} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.3.5

Replace all occurrences of $u$ with ${((1 + u) (1 - u))}^{\frac{1}{2}}$ .

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

Step 4.3.7.3.6

Simplify.

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Step 4.3.7.3.6.1

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

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Step 4.3.7.3.6.1.1

Apply the distributive property.

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

Step 4.3.7.3.6.1.2

Apply the distributive property.

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

Step 4.3.7.3.6.1.3

Apply the distributive property.

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

Step 4.3.7.3.6.2

Simplify and combine like terms.

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Step 4.3.7.3.6.2.1

Simplify each term.

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Step 4.3.7.3.6.2.1.1

Multiply $1$ by $1$ .

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

Step 4.3.7.3.6.2.1.2

Multiply $- u$ by $1$ .

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

Step 4.3.7.3.6.2.1.3

Multiply $u$ by $1$ .

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

Step 4.3.7.3.6.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.7.3.6.2.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.7.3.6.2.1.5.1

Move $u$ .

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

Step 4.3.7.3.6.2.1.5.2

Multiply $u$ by $u$ .

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

Step 4.3.7.3.6.2.2

Add $- u$ and $u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{{(1 + 0 - u^{2})}^{\frac{1}{2}} (1 + {((1 + u) (1 - u))}^{\frac{1}{2}})}{(1 + u) (1 - u)} \cdot (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)} - 1)} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.3.6.2.3

Add $1$ and $0$ .

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

Step 4.3.7.3.6.3

Simplify each term.

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Step 4.3.7.3.6.3.1

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

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Step 4.3.7.3.6.3.1.1

Apply the distributive property.

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

Step 4.3.7.3.6.3.1.2

Apply the distributive property.

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

Step 4.3.7.3.6.3.1.3

Apply the distributive property.

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

Step 4.3.7.3.6.3.2

Simplify and combine like terms.

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Step 4.3.7.3.6.3.2.1

Simplify each term.

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Step 4.3.7.3.6.3.2.1.1

Multiply $1$ by $1$ .

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

Step 4.3.7.3.6.3.2.1.2

Multiply $- u$ by $1$ .

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

Step 4.3.7.3.6.3.2.1.3

Multiply $u$ by $1$ .

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

Step 4.3.7.3.6.3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.7.3.6.3.2.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.7.3.6.3.2.1.5.1

Move $u$ .

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

Step 4.3.7.3.6.3.2.1.5.2

Multiply $u$ by $u$ .

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

Step 4.3.7.3.6.3.2.2

Add $- u$ and $u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{{(1 - u^{2})}^{\frac{1}{2}} (1 + {(1 + 0 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)} \cdot (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)} - 1)} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.3.6.3.2.3

Add $1$ and $0$ .

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

Step 4.3.7.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{(1 + u) (1 - u)}{(1 + u) (1 - u)}$ .

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

Step 4.3.7.5

Combine $- 1$ and $\frac{(1 + u) (1 - u)}{(1 + u) (1 - u)}$ .

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

Step 4.3.7.6

Combine the numerators over the common denominator.

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

Step 4.3.7.7

Rewrite $\frac{\sqrt{(1 + u) (1 - u)} - ((1 + u) (1 - u))}{(1 + u) (1 - u)}$ in a factored form.

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Step 4.3.7.7.1

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

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

Step 4.3.7.7.2

Rewrite $(1 + u) (1 - u)$ as ${({((1 + u) (1 - u))}^{\frac{1}{2}})}^{2}$ .

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

Step 4.3.7.7.3

Let $u = {((1 + u) (1 - u))}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${((1 + u) (1 - u))}^{\frac{1}{2}}$ .

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

Step 4.3.7.7.4

Factor $u$ out of $u - u^{2}$ .

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Step 4.3.7.7.4.1

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

Step 4.3.7.7.4.2

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

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

Step 4.3.7.7.4.3

Factor $u$ out of $- u^{2}$ .

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

Step 4.3.7.7.4.4

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

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

Step 4.3.7.7.5

Replace all occurrences of $u$ with ${((1 + u) (1 - u))}^{\frac{1}{2}}$ .

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

Step 4.3.7.7.6

Simplify.

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Step 4.3.7.7.6.1

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

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Step 4.3.7.7.6.1.1

Apply the distributive property.

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

Step 4.3.7.7.6.1.2

Apply the distributive property.

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

Step 4.3.7.7.6.1.3

Apply the distributive property.

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

Step 4.3.7.7.6.2

Simplify and combine like terms.

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Step 4.3.7.7.6.2.1

Simplify each term.

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Step 4.3.7.7.6.2.1.1

Multiply $1$ by $1$ .

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

Step 4.3.7.7.6.2.1.2

Multiply $- u$ by $1$ .

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

Step 4.3.7.7.6.2.1.3

Multiply $u$ by $1$ .

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

Step 4.3.7.7.6.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.7.7.6.2.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.7.7.6.2.1.5.1

Move $u$ .

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

Step 4.3.7.7.6.2.1.5.2

Multiply $u$ by $u$ .

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

Step 4.3.7.7.6.2.2

Add $- u$ and $u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{{(1 - u^{2})}^{\frac{1}{2}} (1 + {(1 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)} \cdot \frac{{(1 + 0 - u^{2})}^{\frac{1}{2}} (1 - {((1 + u) (1 - u))}^{\frac{1}{2}})}{(1 + u) (1 - u)}} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.7.6.2.3

Add $1$ and $0$ .

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

Step 4.3.7.7.6.3

Simplify each term.

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Step 4.3.7.7.6.3.1

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

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Step 4.3.7.7.6.3.1.1

Apply the distributive property.

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

Step 4.3.7.7.6.3.1.2

Apply the distributive property.

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

Step 4.3.7.7.6.3.1.3

Apply the distributive property.

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

Step 4.3.7.7.6.3.2

Simplify and combine like terms.

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Step 4.3.7.7.6.3.2.1

Simplify each term.

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Step 4.3.7.7.6.3.2.1.1

Multiply $1$ by $1$ .

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

Step 4.3.7.7.6.3.2.1.2

Multiply $- u$ by $1$ .

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

Step 4.3.7.7.6.3.2.1.3

Multiply $u$ by $1$ .

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

Step 4.3.7.7.6.3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.7.7.6.3.2.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.7.7.6.3.2.1.5.1

Move $u$ .

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

Step 4.3.7.7.6.3.2.1.5.2

Multiply $u$ by $u$ .

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

Step 4.3.7.7.6.3.2.2

Add $- u$ and $u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \sqrt{\frac{{(1 - u^{2})}^{\frac{1}{2}} (1 + {(1 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)} \cdot \frac{{(1 - u^{2})}^{\frac{1}{2}} (1 - {(1 + 0 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)}} (\frac{(1 + u) (1 - u)}{\sqrt{(1 + u) (1 - u)}})$

Step 4.3.7.7.6.3.2.3

Add $1$ and $0$ .

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

Step 4.3.8

Multiply $\frac{{(1 - u^{2})}^{\frac{1}{2}} (1 + {(1 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)}$ by $\frac{{(1 - u^{2})}^{\frac{1}{2}} (1 - {(1 - u^{2})}^{\frac{1}{2}})}{(1 + u) (1 - u)}$ .

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

Step 4.3.9

Combine exponents.

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Step 4.3.9.1

Multiply ${(1 - u^{2})}^{\frac{1}{2}}$ by ${(1 - u^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.3.9.1.1

Move ${(1 - u^{2})}^{\frac{1}{2}}$ .

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

Step 4.3.9.1.2

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

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

Step 4.3.9.1.3

Combine the numerators over the common denominator.

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

Step 4.3.9.1.4

Add $1$ and $1$ .

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

Step 4.3.9.1.5

Divide $2$ by $2$ .

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

Step 4.3.9.2

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

Step 4.3.10

Combine exponents.

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Step 4.3.10.1

Raise $1 + u$ to the power of $1$ .

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

Step 4.3.10.2

Raise $1 + u$ to the power of $1$ .

Step 4.3.10.3

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

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

Step 4.3.10.4

Add $1$ and $1$ .

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

Step 4.3.10.5

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

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

Step 4.3.10.6

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

Step 4.3.10.7

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

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

Step 4.3.10.8

Add $1$ and $1$ .

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

Step 4.3.11

Simplify the numerator.

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Step 4.3.11.1

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

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

Step 4.3.11.2

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

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

Step 4.3.12

Factor $1 + u$ out of $(1 + u) (1 - u) (1 + {(1 - u^{2})}^{\frac{1}{2}}) (1 - {(1 - u^{2})}^{\frac{1}{2}})$ .

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

Step 4.3.13

Cancel the common factors.

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Step 4.3.13.1

Factor $1 + u$ out of ${(1 + u)}^{2} {(1 - u)}^{2}$ .

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

Step 4.3.13.2

Cancel the common factor.

Step 4.3.13.3

Rewrite the expression.

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

Step 4.3.14

Factor $1 - u$ out of $((1 - u) (1 + {(1 - u^{2})}^{\frac{1}{2}})) (1 - {(1 - u^{2})}^{\frac{1}{2}})$ .

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

Step 4.3.15

Cancel the common factors.

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Step 4.3.15.1

Factor $1 - u$ out of $(1 + u) {(1 - u)}^{2}$ .

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

Step 4.3.15.2

Cancel the common factor.

Step 4.3.15.3

Rewrite the expression.

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

Step 4.3.16

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

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

Step 4.3.17

Combine.

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

Step 4.3.18

Simplify the denominator.

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Step 4.3.18.1

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

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

Step 4.3.18.2

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

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

Step 4.3.18.3

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

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

Step 4.3.18.4

Add $1$ and $1$ .

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

Step 4.3.19

Simplify the denominator.

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Step 4.3.19.1

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

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Step 4.3.19.1.1

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

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

Step 4.3.19.1.2

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

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

Step 4.3.19.1.3

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

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

Step 4.3.19.1.4

Cancel the common factor of $2$ .

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Step 4.3.19.1.4.1

Cancel the common factor.

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

Step 4.3.19.1.4.2

Rewrite the expression.

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

Step 4.3.19.1.5

Simplify.

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

Step 4.3.19.2

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

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Step 4.3.19.2.1

Apply the distributive property.

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

Step 4.3.19.2.2

Apply the distributive property.

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

Step 4.3.19.2.3

Apply the distributive property.

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

Step 4.3.19.3

Simplify and combine like terms.

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Step 4.3.19.3.1

Simplify each term.

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Step 4.3.19.3.1.1

Multiply $1$ by $1$ .

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

Step 4.3.19.3.1.2

Multiply $- u$ by $1$ .

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

Step 4.3.19.3.1.3

Multiply $u$ by $1$ .

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

Step 4.3.19.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.19.3.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.19.3.1.5.1

Move $u$ .

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

Step 4.3.19.3.1.5.2

Multiply $u$ by $u$ .

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

Step 4.3.19.3.2

Add $- u$ and $u$ .

$f (\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}) = \frac{\sqrt{(1 + {(1 - u^{2})}^{\frac{1}{2}}) (1 - {(1 - u^{2})}^{\frac{1}{2}})} (1 + u) (1 - u)}{1 + 0 - u^{2}}$

Step 4.3.19.3.3

Add $1$ and $0$ .

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

Step 4.3.19.4

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

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

Step 4.3.19.5

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

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

Step 4.3.20

Reduce the expression by cancelling the common factors.

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Step 4.3.20.1

Cancel the common factor.

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

Step 4.3.20.2

Rewrite the expression.

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

Step 4.3.20.3

Cancel the common factor.

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

Step 4.3.20.4

Divide $\sqrt{(1 + {(1 - u^{2})}^{\frac{1}{2}}) (1 - {(1 - u^{2})}^{\frac{1}{2}})}$ by $1$ .

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

Step 4.4

Since $f^{- 1} (f (u)) = u$ and $f (f^{- 1} (u)) = u$ , then $f^{- 1} (u) = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$ is the inverse of $f (u) = \cos (arccsc (u))$ .

$f^{- 1} (u) = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$