Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $(1 - \cos (x)) \sin (x)$ , 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 $\frac{\sin (x)}{\sin (x)}$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Add $1$ and $1$ .

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $- 1$ by $1$ .

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

Step 2.5.4

Multiply $- - \cos (x)$ .

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Step 2.5.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.4.2

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

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

Step 2.5.5

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 2.5.6

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.5.7

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 2.5.8

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 2.5.9

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 2.5.10

Apply pythagorean identity.

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

Step 2.5.11

Factor $\cos (x)$ out of $- \cos^{2} (x) + \cos (x)$ .

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Step 2.5.11.1

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

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

Step 2.5.11.2

Multiply by $1$ .

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

Step 2.5.11.3

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

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

Step 2.6

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

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Step 2.6.1

Reorder terms.

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

Step 2.6.2

Cancel the common factor.

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

Step 2.6.3

Rewrite the expression.

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

Step 3

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

$\cot (x)$

Step 4

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

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