Trigonometry Examples

$\frac{\cos^{2} (x) - 1}{\cos^{2} (x) - \cos (x)}$

Step 1

Rewrite $1$ as $1^{2}$ .

$\frac{\cos^{2} (x) - 1^{2}}{\cos^{2} (x) - \cos (x)}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = 1$ .

$\frac{(\cos (x) + 1) (\cos (x) - 1)}{\cos^{2} (x) - \cos (x)}$

Step 3

Factor $\cos (x)$ out of $\cos^{2} (x) - \cos (x)$ .

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Step 3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 3.2

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 3.3

Factor $\cos (x)$ out of $\cos (x) \cos (x) + \cos (x) \cdot - 1$ .

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

Step 4

Reduce the expression $\frac{(\cos (x) + 1) (\cos (x) - 1)}{\cos (x) (\cos (x) - 1)}$ by cancelling the common factors.

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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