Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Rearrange terms.

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

Step 1.2.2

Apply pythagorean identity.

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

Step 2

Cancel the common factor of $\sin (x) + \cos (x)$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $- \sin (x) \cos (x) + 1$ by $1$ .

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