Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Apply pythagorean identity.

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

Step 4

Multiply $\sec^{2} (x) + \tan^{2} (x)$ by $1$ .

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