Precalculus Examples

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

Step 1

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

$\frac{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}{\sin^{4} (θ) - \cos^{4} (θ)}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

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 $a = \sin^{2} (θ)$ and $b = \cos^{2} (θ)$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply pythagorean identity.

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

Step 2.4.2

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

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

Step 2.4.3

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

$\frac{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}{(\sin (θ) + \cos (θ)) (\sin (θ) - \cos (θ))}$