Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $1 + \cos (x)$ to the power of $1$ .

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

Step 5.2

Raise $1 + \cos (x)$ to the power of $1$ .

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

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

Rewrite $(1 - \cos (x)) (1 - \cos (x))$ as ${(1 - \cos (x))}^{2}$ .

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

Step 5.6

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

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

Step 5.7

Simplify.

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Step 5.7.1

Add $1$ and $1$ .

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

Step 5.7.2

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

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

Step 5.7.3

Add $2$ and $0$ .

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

Step 5.7.4

Apply the distributive property.

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

Step 5.7.5

Multiply $- 1$ by $1$ .

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

Step 5.7.6

Multiply $- - \cos (x)$ .

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Step 5.7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 5.7.6.2

Multiply $\cos (x)$ by $1$ .

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

Step 5.7.7

Subtract $1$ from $1$ .

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

Step 5.7.8

Add $0$ and $\cos (x)$ .

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

Step 5.7.9

Add $\cos (x)$ and $\cos (x)$ .

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

Step 5.7.10

Multiply $2$ by $2$ .

$\frac{4 \cos (x)}{(1 + \cos (x)) (1 - \cos (x))}$