Precalculus Examples

$\cos^{4} (θ) - \sin^{4} (θ) = \cos (2 θ)$

Step 1

Start on the left side.

$\cos^{4} (θ) - \sin^{4} (θ)$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Rewrite $\cos^{4} (θ)$ as ${(\cos^{2} (θ))}^{2}$ .

${(\cos^{2} (θ))}^{2} - \sin^{4} (θ)$

Step 2.2

Rewrite $\sin^{4} (θ)$ as ${(\sin^{2} (θ))}^{2}$ .

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

Step 2.3

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

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

Step 2.4

Rearrange terms.

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

Step 2.5

Apply pythagorean identity.

$1 (\cos^{2} (θ) - \sin^{2} (θ))$

Step 2.6

Multiply $\cos^{2} (θ) - \sin^{2} (θ)$ by $1$ .

$\cos^{2} (θ) - \sin^{2} (θ)$

Step 2.7

Apply the cosine double-angle identity.

$\cos (2 θ)$

Step 3

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

$\cos^{4} (θ) - \sin^{4} (θ) = \cos (2 θ)$ is an identity