Evaluate (1-cos(x))(1+cos(x))

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

Step 1

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

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.

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

Step 2

Simplify and combine like terms.

Simplify each term.

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 $- 1$ by $1$ .

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

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

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 $1$ .

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

Subtract $\cos (x)$ from $\cos (x)$ .

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

Add $1$ and $0$ .

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

Step 3

Apply pythagorean identity.

$\sin^{2} (x)$