Precalculus Examples

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

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

Step 1

Expand

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

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

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Step 1.1

Apply the distributive property.

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

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

Step 1.2

Apply the distributive property.

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

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

Step 1.3

Apply the distributive property.

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

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

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

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 2.1.2

Multiply

\cos (x)

$\cos (x)$ by

1

$1$ .

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

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

Step 2.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.1.4

Multiply

- \cos (x) \cos (x)

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

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Step 2.1.4.1

Raise

\cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.1.4.2

Raise

\cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.1.4.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.1.4.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 2.2

Subtract

\cos (x)

$\cos (x)$ from

\cos (x)

$\cos (x)$ .

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

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

Step 2.3

Add

1

$1$ and

0

$0$ .

1 - \cos^{2} (x)

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

1 - \cos^{2} (x)

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

Step 3

Apply pythagorean identity.

\sin^{2} (x)

$\sin^{2} (x)$