Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Rewrite $\cot (a)$ in terms of sines and cosines.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $\sin (a) \sin (a)$ .

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Step 2.6.1.1

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

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

Step 2.6.1.2

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

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

Step 2.6.1.3

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

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

Step 2.6.1.4

Add $1$ and $1$ .

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Multiply $- 1$ by $1$ .

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

Step 2.6.4

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

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Step 2.6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.4.4

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

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

Step 2.6.4.5

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

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

Step 2.6.4.6

Add $1$ and $1$ .

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

Step 2.6.5

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

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Step 2.6.5.1

Rearrange terms.

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

Step 2.6.5.2

Apply pythagorean identity.

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

Step 2.7

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

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Step 2.7.1

Reorder terms.

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

$\frac{1}{\sin (a)}$

Step 3

Rewrite $\frac{1}{\sin (a)}$ as $\csc (a)$ .

$\csc (a)$

Step 4

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

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