Evaluate (1+cos(theta))(1-cos(theta))=sin(theta)^2

$(1 + \cos (θ)) (1 - \cos (θ)) = \sin^{2} (θ)$

Subtract $\sin^{2} (θ)$ from both sides of the equation.

$(1 + \cos (θ)) (1 - \cos (θ)) - \sin^{2} (θ) = 0$

Simplify $(1 + \cos (θ)) (1 - \cos (θ)) - \sin^{2} (θ)$ .

Simplify each term.

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

Apply the distributive property.

$1 (1 - \cos (θ)) + \cos (θ) (1 - \cos (θ)) - \sin^{2} (θ) = 0$

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (θ)) + \cos (θ) (1 - \cos (θ)) - \sin^{2} (θ) = 0$

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (θ)) + \cos (θ) \cdot 1 + \cos (θ) (- \cos (θ)) - \sin^{2} (θ) = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (- \cos (θ)) + \cos (θ) \cdot 1 + \cos (θ) (- \cos (θ)) - \sin^{2} (θ) = 0$

Multiply $- \cos (θ)$ by $1$ .

$1 - \cos (θ) + \cos (θ) \cdot 1 + \cos (θ) (- \cos (θ)) - \sin^{2} (θ) = 0$

Multiply $\cos (θ)$ by $1$ .

$1 - \cos (θ) + \cos (θ) + \cos (θ) (- \cos (θ)) - \sin^{2} (θ) = 0$

Rewrite using the commutative property of multiplication.

$1 - \cos (θ) + \cos (θ) - \cos (θ) \cos (θ) - \sin^{2} (θ) = 0$

Multiply $- \cos (θ) \cos (θ)$ .

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

$1 - \cos (θ) + \cos (θ) - (\cos^{1} (θ) \cos (θ)) - \sin^{2} (θ) = 0$

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

$1 - \cos (θ) + \cos (θ) - (\cos^{1} (θ) \cos^{1} (θ)) - \sin^{2} (θ) = 0$

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

$1 - \cos (θ) + \cos (θ) - {\cos (θ)}^{1 + 1} - \sin^{2} (θ) = 0$

Add $1$ and $1$ .

$1 - \cos (θ) + \cos (θ) - \cos^{2} (θ) - \sin^{2} (θ) = 0$

Add $- \cos (θ)$ and $\cos (θ)$ .

$1 + 0 - \cos^{2} (θ) - \sin^{2} (θ) = 0$

Add $1$ and $0$ .

$1 - \cos^{2} (θ) - \sin^{2} (θ) = 0$

Apply pythagorean identity.

$\sin^{2} (θ) - \sin^{2} (θ) = 0$

Subtract $\sin^{2} (θ)$ from $\sin^{2} (θ)$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$