Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 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 $b = \cos^{2} (θ)$ .

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

Step 4

Simplify.

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Step 4.1

Apply pythagorean identity.

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

Step 4.2

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

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

Step 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 $b = \cos (θ)$ .

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