Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $\sin (x) (1 - \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $1$ by $1$ .

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

Step 2.5.2.1.2

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

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

Step 2.5.2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.5.2.1.4

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

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Step 2.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2.1.4.2

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

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

Step 2.5.2.1.4.3

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

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

Step 2.5.2.1.4.4

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

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

Step 2.5.2.1.4.5

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

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

Step 2.5.2.1.4.6

Add $1$ and $1$ .

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

Step 2.5.2.2

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

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.3.4

Add $1$ and $1$ .

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

Step 2.5.4

Rewrite $1 - 2 \cos (x) + \cos^{2} (x) + \sin^{2} (x)$ in a factored form.

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Step 2.5.4.1

Rearrange terms.

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

Step 2.5.4.2

Apply pythagorean identity.

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

Step 2.5.4.3

Add $1$ and $1$ .

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

Step 2.5.4.4

Factor $2$ out of $2 - 2 \cos (x)$ .

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Step 2.5.4.4.1

Factor $2$ out of $2$ .

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

Step 2.5.4.4.2

Factor $2$ out of $- 2 \cos (x)$ .

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

Step 2.5.4.4.3

Factor $2$ out of $2 (1) + 2 (- \cos (x))$ .

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

Step 2.6

Cancel the common factor of $1 - \cos (x)$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 3

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

$2 \csc (x)$

Step 4

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

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