Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.1.2

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

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

Step 1.2

Cancel the common factor of $\sec (x) + 3$ and $3 + \sec (x)$ .

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Step 1.2.1

Reorder terms.

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Divide $\sec (x) - 3$ by $1$ .

$\sec (x) - 3 - \sec (x)$

Step 2

Combine the opposite terms in $\sec (x) - 3 - \sec (x)$ .

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Step 2.1

Subtract $\sec (x)$ from $\sec (x)$ .

$0 - 3$

Step 2.2

Subtract $3$ from $0$ .

$- 3$