Precalculus Examples

$\frac{\cos^{3} (x) - \sin^{3} (x)}{\sin (x) - \cos (x)}$

Step 1

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = \cos (x)$ and $b = \sin (x)$ .

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

Step 2

Simplify.

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Step 2.1

Rearrange terms.

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

Step 2.2

Rearrange terms.

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

Step 2.3

Apply pythagorean identity.

$\frac{(\cos (x) - \sin (x)) (\cos (x) \sin (x) + 1)}{\sin (x) - \cos (x)}$