Precalculus Examples

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

Step 1

To write $\frac{1}{1 - \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \cos (x)}{1 + \cos (x)}$ .

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

Step 2

To write $\frac{1}{1 + \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cos (x)}{1 - \cos (x)}$ .

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

Step 3

Write each expression with a common denominator of $(1 - \cos (x)) (1 + \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(1 - \cos (x)) (1 + \cos (x))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Add $1$ and $1$ .

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

Step 5.2

Subtract $\cos (x)$ from $\cos (x)$ .

$\frac{2 + 0}{(1 + \cos (x)) (1 - \cos (x))}$

Step 5.3

Add $2$ and $0$ .

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