Precalculus Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$\sqrt{1 + x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Multiply $- 1$ by $- 1$ .

$\sqrt{1 + x^{2} - x - x + 1}$

Step 3.2

Subtract $x$ from $- x$ .

$\sqrt{1 + x^{2} - 2 x + 1}$

Step 4

Add $1$ and $1$ .

$\sqrt{x^{2} - 2 x + 2}$