Precalculus Examples

\sec (\arcsin (x - 1))

$\sec (\arcsin (x - 1))$

Step 1

Draw a triangle in the plane with vertices

(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)

$(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)$ ,

(\sqrt{1^{2} - {(x - 1)}^{2}}, 0)

$(\sqrt{1^{2} - {(x - 1)}^{2}}, 0)$ , and the origin. Then

\arcsin (x - 1)

$\arcsin (x - 1)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)

$(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)$ . Therefore,

\sec (\arcsin (x - 1))

$\sec (\arcsin (x - 1))$ is

\frac{1}{\sqrt{1 - {(x - 1)}^{2}}}

$\frac{1}{\sqrt{1 - {(x - 1)}^{2}}}$ .

\frac{1}{\sqrt{1 - {(x - 1)}^{2}}}

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\frac{1}{\sqrt{1^{2} - {(x - 1)}^{2}}}

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = x - 1

$b = x - 1$ .

\frac{1}{\sqrt{(1 + x - 1) (1 - (x - 1))}}

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

Step 2.3

Simplify.

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Step 2.3.1

Subtract

1

$1$ from

1

$1$ .

\frac{1}{\sqrt{(x + 0) (1 - (x - 1))}}

$\frac{1}{\sqrt{(x + 0) (1 - (x - 1))}}$

Step 2.3.2

Add

x

$x$ and

0

$0$ .

\frac{1}{\sqrt{x (1 - (x - 1))}}

$\frac{1}{\sqrt{x (1 - (x - 1))}}$

Step 2.3.3

Apply the distributive property.

\frac{1}{\sqrt{x (1 - x - - 1)}}

$\frac{1}{\sqrt{x (1 - x - - 1)}}$

Step 2.3.4

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{1}{\sqrt{x (1 - x + 1)}}

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

Step 2.3.5

Add

1

$1$ and

1

$1$ .

\frac{1}{\sqrt{x (- x + 2)}}

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

\frac{1}{\sqrt{x (- x + 2)}}

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

\frac{1}{\sqrt{x (- x + 2)}}

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

Step 3

Multiply

\frac{1}{\sqrt{x (- x + 2)}}

$\frac{1}{\sqrt{x (- x + 2)}}$ by

\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}

$\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}$ .

\frac{1}{\sqrt{x (- x + 2)}} \cdot \frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

\frac{1}{\sqrt{x (- x + 2)}}

$\frac{1}{\sqrt{x (- x + 2)}}$ by

\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}

$\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}$ .

\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)} \sqrt{x (- x + 2)}}

$\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)} \sqrt{x (- x + 2)}}$

Step 4.2

Raise

\sqrt{x (- x + 2)}

$\sqrt{x (- x + 2)}$ to the power of

1

$1$ .

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

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

Step 4.3

Raise

\sqrt{x (- x + 2)}

$\sqrt{x (- x + 2)}$ to the power of

1

$1$ .

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

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

Step 4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{x (- x + 2)}}{{\sqrt{x (- x + 2)}}^{2}}

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

Step 4.6

Rewrite

{\sqrt{x (- x + 2)}}^{2}

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

x (- x + 2)

$x (- x + 2)$ .

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Step 4.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x (- x + 2)}

$\sqrt{x (- x + 2)}$ as

{(x (- x + 2))}^{\frac{1}{2}}

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

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

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

Step 4.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{x (- x + 2)}}{{(x (- x + 2))}^{\frac{2}{2}}}

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

Step 4.6.4

Cancel the common factor of

2

$2$ .

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Step 4.6.4.1

Cancel the common factor.

\frac{\sqrt{x (- x + 2)}}{{(x (- x + 2))}^{\frac{2}{2}}}

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

Step 4.6.4.2

Rewrite the expression.

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

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

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

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

Step 4.6.5

Simplify.

\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}

$\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}$

\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}

$\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}$

\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}

$\frac{\sqrt{x (- x + 2)}}{x (- x + 2)}$

Step 5

Factor

- 1

$- 1$ out of

- x

$- x$ .

\frac{\sqrt{x (- x + 2)}}{x (- (x) + 2)}

$\frac{\sqrt{x (- x + 2)}}{x (- (x) + 2)}$

Step 6

Rewrite

2

$2$ as

- 1 (- 2)

$- 1 (- 2)$ .

\frac{\sqrt{x (- x + 2)}}{x (- (x) - 1 (- 2))}

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

Step 7

Factor

- 1

$- 1$ out of

- (x) - 1 (- 2)

$- (x) - 1 (- 2)$ .

\frac{\sqrt{x (- x + 2)}}{x (- (x - 2))}

$\frac{\sqrt{x (- x + 2)}}{x (- (x - 2))}$

Step 8

Rewrite negatives.

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Step 8.1

Rewrite

- (x - 2)

$- (x - 2)$ as

- 1 (x - 2)

$- 1 (x - 2)$ .

\frac{\sqrt{x (- x + 2)}}{x (- 1 (x - 2))}

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

Step 8.2

Move the negative in front of the fraction.

- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}

$- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}$

- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}

$- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}$