Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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 $b = \tan^{2} (x)$ .

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

Step 1.4

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 $b = \tan (x)$ .

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

Step 2

Cancel the common factor of $\sec^{2} (x) + \tan^{2} (x)$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ by $1$ .

$(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$

Step 3

Expand $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sec (x) (\sec (x) - \tan (x)) + \tan (x) (\sec (x) - \tan (x))$

Step 3.2

Apply the distributive property.

$\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) (\sec (x) - \tan (x))$

Step 3.3

Apply the distributive property.

$\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) \sec (x) + \tan (x) (- \tan (x))$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) \sec (x) + \tan (x) (- \tan (x))$ .

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Step 4.1.1

Reorder the factors in the terms $\sec (x) (- \tan (x))$ and $\tan (x) \sec (x)$ .

$\sec (x) \sec (x) - \sec (x) \tan (x) + \sec (x) \tan (x) + \tan (x) (- \tan (x))$

Step 4.1.2

Add $- \sec (x) \tan (x)$ and $\sec (x) \tan (x)$ .

$\sec (x) \sec (x) + 0 + \tan (x) (- \tan (x))$

Step 4.1.3

Add $\sec (x) \sec (x)$ and $0$ .

$\sec (x) \sec (x) + \tan (x) (- \tan (x))$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $\sec (x) \sec (x)$ .

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Step 4.2.1.1

Raise $\sec (x)$ to the power of $1$ .

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

Step 4.2.1.2

Raise $\sec (x)$ to the power of $1$ .

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

Step 4.2.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.1.4

Add $1$ and $1$ .

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

Step 4.2.2

Rewrite using the commutative property of multiplication.

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

Step 4.2.3

Multiply $- \tan (x) \tan (x)$ .

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Step 4.2.3.1

Raise $\tan (x)$ to the power of $1$ .

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

Step 4.2.3.2

Raise $\tan (x)$ to the power of $1$ .

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

Step 4.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.3.4

Add $1$ and $1$ .

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

Step 5

Apply pythagorean identity.

$1$