Precalculus Examples

$\frac{1 - \cos^{4} (x)}{1 + \cos^{2} (x)}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - \cos^{4} (x)}{1 + \cos^{2} (x)}$

Step 1.2

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

$\frac{1^{2} - {(\cos^{2} (x))}^{2}}{1 + \cos^{2} (x)}$

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 $a = 1$ and $b = \cos^{2} (x)$ .

$\frac{(1 + \cos^{2} (x)) (1 - \cos^{2} (x))}{1 + \cos^{2} (x)}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(1 + \cos^{2} (x)) (1^{2} - \cos^{2} (x))}{1 + \cos^{2} (x)}$

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

$\frac{(1 + \cos^{2} (x)) (1 + \cos (x)) (1 - \cos (x))}{1 + \cos^{2} (x)}$

Step 2

Cancel the common factor of $1 + \cos^{2} (x)$ .

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Step 2.1

Cancel the common factor.

$\frac{(1 + \cos^{2} (x)) (1 + \cos (x)) (1 - \cos (x))}{1 + \cos^{2} (x)}$

Step 2.2

Divide $(1 + \cos (x)) (1 - \cos (x))$ by $1$ .

$(1 + \cos (x)) (1 - \cos (x))$

Step 3

Expand $(1 + \cos (x)) (1 - \cos (x))$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$1 (1 - \cos (x)) + \cos (x) (1 - \cos (x))$

Step 3.2

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (x)) + \cos (x) (1 - \cos (x))$

Step 3.3

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $1$ by $1$ .

$1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))$

Step 4.1.2

Multiply $- \cos (x)$ by $1$ .

$1 - \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))$

Step 4.1.3

Multiply $\cos (x)$ by $1$ .

$1 - \cos (x) + \cos (x) + \cos (x) (- \cos (x))$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$1 - \cos (x) + \cos (x) - \cos (x) \cos (x)$

Step 4.1.5

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

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Step 4.1.5.1

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

$1 - \cos (x) + \cos (x) - (\cos^{1} (x) \cos (x))$

Step 4.1.5.2

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

$1 - \cos (x) + \cos (x) - (\cos^{1} (x) \cos^{1} (x))$

Step 4.1.5.3

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

$1 - \cos (x) + \cos (x) - {\cos (x)}^{1 + 1}$

Step 4.1.5.4

Add $1$ and $1$ .

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

Step 4.2

Add $- \cos (x)$ and $\cos (x)$ .

$1 + 0 - \cos^{2} (x)$

Step 4.3

Add $1$ and $0$ .

$1 - \cos^{2} (x)$

Step 5

Apply pythagorean identity.

$\sin^{2} (x)$