Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Multiply $- - \cos (x)$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.2

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

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

Step 5.5

Subtract $1$ from $1$ .

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

Step 5.6

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

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

Step 5.7

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

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

Step 6

Move $2$ to the left of $\sin (x)$ .

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