Simplify (1+cos(x))(1-cos(x))

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

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

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

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

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

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $1$ and $0$ .

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

Apply pythagorean identity.

$\sin^{2} (x)$

Simplify (1+cos(x))(1-cos(x))

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