Trigonometry Examples

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

Regroup terms.

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

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

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

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 $b = 1$ .

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

Simplify.

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Multiply $- 1$ by $1$ .

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

One to any power is one.

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

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

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

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

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

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

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

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

Factor.

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

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

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

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

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

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

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

Remove unnecessary parentheses.

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

Factor out negative.

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

Remove parentheses.

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

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

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

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

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

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

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

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

Factor using the perfect square rule.

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Rearrange terms.

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

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

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

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

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

Rewrite the polynomial.

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

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

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