Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Add fractions.

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Step 2.1

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 2.2

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 2.3

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 2.3.1

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 2.3.2

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 2.3.3

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 2.4

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 3

Simplify numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Reorder $\sin (x)$ and $- 1$ .

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

Step 3.3.2

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

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

Step 3.4

Add $2 \sin (x)$ and $0$ .

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

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

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

Step 6

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

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

Step 7

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

$2 \csc (x)$

Step 8

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

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