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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) \cdot 1 - 1 \cos (x) - 1 \cdot 1$ .

Reorder the factors in the terms $\cos (x) \cdot 1$ and $- 1 \cos (x)$ .

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

Subtract $1 \cos (x)$ from $1 \cos (x)$ .

$\cos (x) \cos (x) + 0 - 1 \cdot 1$

Add $\cos (x) \cos (x)$ and $0$ .

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

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $- 1$ by $1$ .

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

Simplify with factoring out.

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

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

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

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

Apply pythagorean identity.

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