Trigonometry Examples

$\sec^{4} (x) - 2 \sec^{2} (x) \tan^{2} (x) + \tan^{4} (x)$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $\sec^{4} (x)$ as ${(\sec^{2} (x))}^{2}$ .

${(\sec^{2} (x))}^{2} - 2 \sec^{2} (x) \tan^{2} (x) + \tan^{4} (x)$

Step 1.2

Rewrite $\tan^{4} (x)$ as ${(\tan^{2} (x))}^{2}$ .

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

Step 1.3

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

$2 \sec^{2} (x) \tan^{2} (x) = 2 \cdot \sec^{2} (x) \cdot \tan^{2} (x)$

Step 1.4

Rewrite the polynomial.

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

Step 1.5

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

${(\sec^{2} (x) - \tan^{2} (x))}^{2}$

Step 2

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

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

Step 3

Apply the product rule to $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ .

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