Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Factor.

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Step 4.1

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 $b = \tan (x)$ .

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

Step 4.2

Remove unnecessary parentheses.

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