Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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Step 6.2.1

Combine $\sin (a)$ and $\frac{\sin (a)}{\cos (a)}$ .

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.2.5

Add $1$ and $1$ .

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

Step 6.3

Move $- 1$ to the left of $\sin (a)$ .

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

Step 6.4

Rewrite $- 1 \sin (a)$ as $- \sin (a)$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Multiply $- 1$ by $1$ .

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

Step 6.7

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

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Step 6.7.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.2

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

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

Step 6.7.3

Combine $\cos (a)$ and $\frac{\cos (a)}{\sin (a)}$ .

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

Step 6.7.4

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

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

Step 6.7.5

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

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

Step 6.7.6

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

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

Step 6.7.7

Add $1$ and $1$ .

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

Step 7

Simplify each term.

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Step 7.1

Factor $\sin (a)$ out of $\sin^{2} (a)$ .

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

Step 7.2

Separate fractions.

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

Step 7.3

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

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

Step 7.4

Divide $\sin (a)$ by $1$ .

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

Step 7.5

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

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

Step 7.6

Separate fractions.

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

Step 7.7

Convert from $\frac{\cos (a)}{\sin (a)}$ to $\cot (a)$ .

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

Step 7.8

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

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

Step 8

Simplify the denominator.

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Step 8.1

Convert from $\frac{\cos (a)}{\sin (a)}$ to $\cot (a)$ .

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

Step 8.2

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

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