Precalculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Add $1$ and $1$ .

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

Step 3

Apply the cosine double-angle identity.

$\cos (2 x)$