Calculus Examples

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

Step 1

Write the expression using exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{3}$ and $b = 1$ .

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

Step 3

Combine into one fraction.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{x + 3}{3} \cdot \frac{x - 1 \cdot 3}{3}}$

Step 7

Multiply $- 1$ by $3$ .

$\sqrt{\frac{x + 3}{3} \cdot \frac{x - 3}{3}}$

Step 8

Multiply $\frac{x + 3}{3}$ by $\frac{x - 3}{3}$ .

$\sqrt{\frac{(x + 3) (x - 3)}{3 \cdot 3}}$

Step 9

Multiply $3$ by $3$ .

$\sqrt{\frac{(x + 3) (x - 3)}{9}}$

Step 10

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

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Step 10.1

Factor the perfect power $1^{2}$ out of $(x + 3) (x - 3)$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

Combine $\frac{1}{3}$ and $\sqrt{(x + 3) (x - 3)}$ .

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