Trigonometry 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

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $1$ .

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

Step 5.3

Subtract $1$ from $1$ .

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

Step 5.4

Subtract $\cos (x)$ from $0$ .

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

Step 5.5

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

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

Step 6

Move the negative in front of the fraction.

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