Calculus Examples

$\sec (\sin^{-1} (u))$

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 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 $b = u$ .

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 6

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 7

Substitute the actual values of $a = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$ and $b = 0$ .

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

Step 8

Find $| z |$ .

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Step 8.1

Raising $0$ to any positive power yields $0$ .

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

Step 8.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.1

Apply the product rule to $\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$ .

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

Step 8.2.2

Apply the product rule to $(1 + u) (1 - u)$ .

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

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Step 8.3.1.1

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

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

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

Step 8.3.1.4

Cancel the common factor of $2$ .

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Step 8.3.1.4.1

Cancel the common factor.

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

Step 8.3.1.4.2

Rewrite the expression.

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

Step 8.3.1.5

Simplify.

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

Step 8.3.2

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

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Step 8.3.2.1

Apply the distributive property.

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

Step 8.3.2.2

Apply the distributive property.

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

Step 8.3.2.3

Apply the distributive property.

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

Step 8.3.3

Simplify and combine like terms.

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Step 8.3.3.1

Simplify each term.

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Step 8.3.3.1.1

Multiply $1$ by $1$ .

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

Step 8.3.3.1.2

Multiply $- u$ by $1$ .

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

Step 8.3.3.1.3

Multiply $u$ by $1$ .

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

Step 8.3.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 8.3.3.1.5

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

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Step 8.3.3.1.5.1

Move $u$ .

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

Step 8.3.3.1.5.2

Multiply $u$ by $u$ .

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

Step 8.3.3.2

Add $- u$ and $u$ .

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

Step 8.3.3.3

Add $1$ and $0$ .

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

Step 8.3.4

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

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

Step 8.3.5

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 $b = u$ .

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

Step 8.4

Cancel the common factors.

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Step 8.4.1

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

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 8.5

Cancel the common factor of $1 - u$ and ${(1 - u)}^{2}$ .

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Step 8.5.1

Multiply by $1$ .

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

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

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

$| z | = \sqrt{0 + \frac{(1 - u) \cdot 1}{(1 - u) ((1 + u) (1 - u))}}$

Step 8.5.2.2

Cancel the common factor.

$| z | = \sqrt{0 + \frac{(1 - u) \cdot 1}{(1 - u) ((1 + u) (1 - u))}}$

Step 8.5.2.3

Rewrite the expression.

$| z | = \sqrt{0 + \frac{1}{(1 + u) (1 - u)}}$

Step 8.6

Add $0$ and $\frac{1}{(1 + u) (1 - u)}$ .

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

Step 8.7

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

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

Step 8.8

Any root of $1$ is $1$ .

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

Step 8.9

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

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

Step 8.10

Combine and simplify the denominator.

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Step 8.10.1

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

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

Step 8.10.2

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

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

Step 8.10.3

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

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

Step 8.10.4

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

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

Step 8.10.5

Add $1$ and $1$ .

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

Step 8.10.6

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

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Step 8.10.6.1

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

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

Step 8.10.6.2

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

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

Step 8.10.6.3

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

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

Step 8.10.6.4

Cancel the common factor of $2$ .

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Step 8.10.6.4.1

Cancel the common factor.

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

Step 8.10.6.4.2

Rewrite the expression.

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

Step 8.10.6.5

Simplify.

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

Step 9

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \tan^{-1} (\frac{0}{\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}})$

Step 10

Substitute the values of $θ = \tan^{-1} (\frac{0}{\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}})$ and $| z | = \frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$ .

$\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u) (\cos (\tan^{-1} (\frac{0}{\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}})) + i \sin (\tan^{-1} (\frac{0}{\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}})))}$