Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify with factoring out.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Apply pythagorean identity.

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

Step 5

Simplify terms.

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Step 5.1

Multiply $- 1$ by $1$ .

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

Step 5.2

Simplify by multiplying through.

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Reorder.

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Step 5.2.2.1

Move $- 1$ to the left of $\tan^{2} (x)$ .

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

Step 5.2.2.2

Move $- 1$ to the left of $\sec^{2} (x)$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Rewrite $- 1 \tan^{2} (x)$ as $- \tan^{2} (x)$ .

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

Step 5.3.2

Rewrite $- 1 \sec^{2} (x)$ as $- \sec^{2} (x)$ .

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