Trigonometry Examples

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

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 $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 2

Simplify.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

$(\sin (x) - \cos (x)) (\sin (x) \cos (x) + 1)$