Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Add $1$ and $1$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 6.4

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

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

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

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

Step 6.4.5

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

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

Step 6.4.6

Add $1$ and $1$ .

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

Step 6.5

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

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Step 6.5.1

Rearrange terms.

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

Step 6.5.2

Rearrange terms.

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

Step 6.5.3

Apply pythagorean identity.

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

Step 7

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

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Step 7.1

Reorder terms.

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x)$