Simplify (1-cos(2x))(1+cos(2x))

$(1 - \cos (2 x)) (1 + \cos (2 x))$

Expand $(1 - \cos (2 x)) (1 + \cos (2 x))$ using the FOIL Method.

Apply the distributive property.

$1 (1 + \cos (2 x)) - \cos (2 x) (1 + \cos (2 x))$

Apply the distributive property.

$1 \cdot 1 + 1 \cos (2 x) - \cos (2 x) (1 + \cos (2 x))$

Apply the distributive property.

$1 \cdot 1 + 1 \cos (2 x) - \cos (2 x) \cdot 1 - \cos (2 x) \cos (2 x)$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 \cos (2 x) - \cos (2 x) \cdot 1 - \cos (2 x) \cos (2 x)$

Multiply $\cos (2 x)$ by $1$ .

$1 + \cos (2 x) - \cos (2 x) \cdot 1 - \cos (2 x) \cos (2 x)$

Multiply $- 1$ by $1$ .

$1 + \cos (2 x) - \cos (2 x) - \cos (2 x) \cos (2 x)$

Multiply $- \cos (2 x) \cos (2 x)$ .

Raise $\cos (2 x)$ to the power of $1$ .

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

Raise $\cos (2 x)$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Subtract $\cos (2 x)$ from $\cos (2 x)$ .

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

Add $1$ and $0$ .

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

Apply pythagorean identity.

$\sin^{2} (2 x)$