Trigonometry Examples

$\frac{\cos^{3} (A) - 8}{\cos (A) - 2}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

$\frac{\cos^{3} (A) - 2^{3}}{\cos (A) - 2}$

Step 1.2

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 $a = \cos (A)$ and $b = 2$ .

$\frac{(\cos (A) - 2) (\cos^{2} (A) + \cos (A) \cdot 2 + 2^{2})}{\cos (A) - 2}$

Step 1.3

Simplify.

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Step 1.3.1

Move $2$ to the left of $\cos (A)$ .

$\frac{(\cos (A) - 2) (\cos^{2} (A) + 2 \cdot \cos (A) + 2^{2})}{\cos (A) - 2}$

Step 1.3.2

Raise $2$ to the power of $2$ .

$\frac{(\cos (A) - 2) (\cos^{2} (A) + 2 \cos (A) + 4)}{\cos (A) - 2}$

Step 2

Cancel the common factor of $\cos (A) - 2$ .

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Step 2.1

Cancel the common factor.

$\frac{(\cos (A) - 2) (\cos^{2} (A) + 2 \cos (A) + 4)}{\cos (A) - 2}$

Step 2.2

Divide $\cos^{2} (A) + 2 \cos (A) + 4$ by $1$ .

$\cos^{2} (A) + 2 \cos (A) + 4$