Precalculus Examples

$\cos (2 a) = \cos^{2} (a) - \sin^{2} (a)$

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Simplify each term.

Tap for more steps...

Step 2.4.1

Multiply $\cos (a) \cos (a)$ .

Tap for more steps...

Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.1.3

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

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

Step 2.4.1.4

Add $1$ and $1$ .

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

Step 2.4.2

Rewrite using the commutative property of multiplication.

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

Step 2.4.3

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

Tap for more steps...

Step 2.4.3.1

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

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

Step 2.4.3.2

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

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

Step 2.4.3.3

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

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

Step 2.4.3.4

Add $1$ and $1$ .

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

Step 2.5

Apply the cosine double-angle identity.

$\cos (2 a)$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos (2 a) = \cos^{2} (a) - \sin^{2} (a)$ is an identity