Precalculus Examples

$\sin^{4} (θ) - \cos^{4} (θ)$

Step 1

Simplify the expression.

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Step 1.1

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

${(\sin^{2} (θ))}^{2} - \cos^{4} (θ)$

Step 1.2

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

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

Step 2

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

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

Step 3

Apply pythagorean identity.

$1 (\sin^{2} (θ) - \cos^{2} (θ))$

Step 4

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

$\sin^{2} (θ) - \cos^{2} (θ)$