Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Add fractions.

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Step 2.1

To write

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

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

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

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

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

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

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

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

\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)}

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

Step 2.3

Write each expression with a common denominator of

(1 - \cos (x)) (1 + \cos (x))

$(1 - \cos (x)) (1 + \cos (x))$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 2.3.1

Multiply

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

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

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

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

\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)}

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

Step 2.3.2

Multiply

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

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

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

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

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

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

Step 2.3.3

Reorder the factors of

(1 - \cos (x)) (1 + \cos (x))

$(1 - \cos (x)) (1 + \cos (x))$ .

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

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

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

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

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

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

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))}

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

Step 3.1.2

Multiply

\sin (x)

$\sin (x)$ by

1

$1$ .

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

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

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))}

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

Step 3.1.4

Multiply

\sin (x)

$\sin (x)$ by

1

$1$ .

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

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

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

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

Step 3.2

Add

\sin (x)

$\sin (x)$ and

\sin (x)

$\sin (x)$ .

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

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

\sin (x) (- \cos (x))

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

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Step 3.3.1

Reorder

\sin (x)

$\sin (x)$ and

- 1

$- 1$ .

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

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

Step 3.3.2

Subtract

\sin (x) \cos (x)

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

\sin (x) \cos (x)

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

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

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

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

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

Step 3.4

Add

2 \sin (x)

$2 \sin (x)$ and

0

$0$ .

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

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

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

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

Step 4

Simplify denominator.

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Step 4.1

Expand

(1 + \cos (x)) (1 - \cos (x))

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

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

Step 4.1.2

Apply the distributive property.

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

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

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))}

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

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

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

Step 4.2

Simplify and combine like terms.

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

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

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

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

Step 5

Apply pythagorean identity.

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

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

Step 6

Cancel the common factor of

\sin (x)

$\sin (x)$ and

{\sin (x)}^{2}

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

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

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

Step 7

Rewrite

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

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

2 \csc (x)

$2 \csc (x)$ .

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)

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