Trigonometry Examples

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

Step 1

Factor $\cos (t) \sin (t)$ out of $\cos^{3} (t) \sin (t) - \sin^{3} (t) \cos (t)$ .

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Step 1.1

Factor $\cos (t) \sin (t)$ out of $\cos^{3} (t) \sin (t)$ .

$\cos (t) \sin (t) (\cos^{2} (t)) - \sin^{3} (t) \cos (t)$

Step 1.2

Factor $\cos (t) \sin (t)$ out of $- \sin^{3} (t) \cos (t)$ .

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

Step 1.3

Factor $\cos (t) \sin (t)$ out of $\cos (t) \sin (t) (\cos^{2} (t)) + \cos (t) \sin (t) (- \sin^{2} (t))$ .

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

Step 2

Factor.

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Step 2.1

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

$\cos (t) \sin (t) ((\cos (t) + \sin (t)) (\cos (t) - \sin (t)))$

Step 2.2

Remove unnecessary parentheses.

$\cos (t) \sin (t) (\cos (t) + \sin (t)) (\cos (t) - \sin (t))$