Evaluate cos(x)^2-sin(x)^2

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

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

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

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

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

$\cos (x) \cos (x) + 0 + \sin (x) (- \sin (x))$

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

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

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Apply the cosine double-angle identity.

$\cos (2 x)$

Evaluate cos(x)^2-sin(x)^2

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