Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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

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

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

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

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

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

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

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

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

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

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 3

Simplify numerator.

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Simplify each term.

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Apply the distributive property.

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Multiply $- \sin (x) (- \cos (x))$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Step 4

Simplify denominator.

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Expand $(1 + \cos (x)) (1 - \cos (x))$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

$\frac{2 \sin (x) \cos (x)}{\sin^{2} (x)}$

Step 6

Cancel the common factor of $\sin (x)$ and ${\sin (x)}^{2}$ .

$\frac{2 \cos (x)}{\sin (x)}$

Step 7

Rewrite $\frac{2 \cos (x)}{\sin (x)}$ as $2 \cot (x)$ .

$2 \cot (x)$

Step 8

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sin (x)}{1 - \cos (x)} - \frac{\sin (x)}{1 + \cos (x)} = 2 \cot (x)$ is an identity