Trigonometry Examples

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

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

$a = \sin (θ)$ and

$b = \cos (θ)$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Rearrange terms.

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

Step 1.2.2

Apply pythagorean identity.

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

Step 2

Cancel the common factor of

$\sin (θ) + \cos (θ)$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide

$- \sin (θ) \cos (θ) + 1$ by

$1$ .

$- \sin (θ) \cos (θ) + 1$