Trigonometry Examples

$5 \sec^{3} (x) - 5 \sec^{2} (x) - 5 \sec (x) + 5$

Step 1

Factor $5$ out of $5 \sec^{3} (x) - 5 \sec^{2} (x) - 5 \sec (x) + 5$ .

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Step 1.1

Factor $5$ out of $5 \sec^{3} (x)$ .

$5 (\sec^{3} (x)) - 5 \sec^{2} (x) - 5 \sec (x) + 5$

Step 1.2

Factor $5$ out of $- 5 \sec^{2} (x)$ .

$5 (\sec^{3} (x)) + 5 (- \sec^{2} (x)) - 5 \sec (x) + 5$

Step 1.3

Factor $5$ out of $- 5 \sec (x)$ .

$5 (\sec^{3} (x)) + 5 (- \sec^{2} (x)) + 5 (- \sec (x)) + 5$

Step 1.4

Factor $5$ out of $5$ .

$5 (\sec^{3} (x)) + 5 (- \sec^{2} (x)) + 5 (- \sec (x)) + 5 (1)$

Step 1.5

Factor $5$ out of $5 (\sec^{3} (x)) + 5 (- \sec^{2} (x))$ .

$5 (\sec^{3} (x) - \sec^{2} (x)) + 5 (- \sec (x)) + 5 (1)$

Step 1.6

Factor $5$ out of $5 (\sec^{3} (x) - \sec^{2} (x)) + 5 (- \sec (x))$ .

$5 (\sec^{3} (x) - \sec^{2} (x) - \sec (x)) + 5 (1)$

Step 1.7

Factor $5$ out of $5 (\sec^{3} (x) - \sec^{2} (x) - \sec (x)) + 5 (1)$ .

$5 (\sec^{3} (x) - \sec^{2} (x) - \sec (x) + 1)$

Step 2

Regroup terms.

$5 (\sec^{3} (x) + 1 - \sec^{2} (x) - \sec (x))$

Step 3

Rewrite $1$ as $1^{3}$ .

$5 (\sec^{3} (x) + 1^{3} - \sec^{2} (x) - \sec (x))$

Step 4

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

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) \cdot 1 + 1^{2}) - \sec^{2} (x) - \sec (x))$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $1$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1^{2}) - \sec^{2} (x) - \sec (x))$

Step 5.2

One to any power is one.

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) - \sec^{2} (x) - \sec (x))$

Step 6

Factor $\sec (x)$ out of $- \sec^{2} (x) - \sec (x)$ .

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Step 6.1

Factor $\sec (x)$ out of $- \sec^{2} (x)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- \sec (x)) - \sec (x))$

Step 6.2

Factor $\sec (x)$ out of $- \sec (x)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- \sec (x)) + \sec (x) \cdot - 1)$

Step 6.3

Factor $\sec (x)$ out of $\sec (x) (- \sec (x)) + \sec (x) \cdot - 1$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- \sec (x) - 1))$

Step 7

Factor.

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Step 7.1

Factor $- 1$ out of $- \sec (x) - 1$ .

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Step 7.1.1

Factor $- 1$ out of $- \sec (x)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- (\sec (x)) - 1))$

Step 7.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- (\sec (x)) - 1 (1)))$

Step 7.1.3

Factor $- 1$ out of $- (\sec (x)) - 1 (1)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) (- (\sec (x) + 1)))$

Step 7.2

Remove unnecessary parentheses.

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + \sec (x) \cdot - 1 (\sec (x) + 1))$

Step 8

Factor out negative.

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) - (\sec (x) (\sec (x) + 1)))$

Step 9

Remove parentheses.

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) - \sec (x) (\sec (x) + 1))$

Step 10

Factor $\sec (x) + 1$ out of $(\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) - \sec (x) (\sec (x) + 1)$ .

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Step 10.1

Factor $\sec (x) + 1$ out of $- \sec (x) (\sec (x) + 1)$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + (\sec (x) + 1) (- \sec (x)))$

Step 10.2

Factor $\sec (x) + 1$ out of $(\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1) + (\sec (x) + 1) (- \sec (x))$ .

$5 ((\sec (x) + 1) (\sec^{2} (x) - \sec (x) + 1 - \sec (x)))$

Step 11

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

$5 ((\sec (x) + 1) (\sec^{2} (x) + 1 - 2 \sec (x)))$

Step 12

Factor.

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Step 12.1

Factor using the perfect square rule.

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Step 12.1.1

Rearrange terms.

$5 ((\sec (x) + 1) (\sec^{2} (x) - 2 \sec (x) + 1))$

Step 12.1.2

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

$5 ((\sec (x) + 1) (\sec^{2} (x) - 2 \sec (x) + 1^{2}))$

Step 12.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sec (x) = 2 \cdot \sec (x) \cdot 1$

Step 12.1.4

Rewrite the polynomial.

$5 ((\sec (x) + 1) (\sec^{2} (x) - 2 \cdot \sec (x) \cdot 1 + 1^{2}))$

Step 12.1.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = \sec (x)$ and $b = 1$ .

$5 ((\sec (x) + 1) {(\sec (x) - 1)}^{2})$

Step 12.2

Remove unnecessary parentheses.

$5 (\sec (x) + 1) {(\sec (x) - 1)}^{2}$