Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Multiply $1$ by $1$ .

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

Step 5.3.1.2

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

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

Step 5.3.1.3

Multiply $- 1$ by $1$ .

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

Step 5.3.1.4

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

Tap for more steps...

Step 5.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.1.4.2

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

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

Step 5.3.1.4.3

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

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

Step 5.3.1.4.4

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

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

Step 5.3.1.4.5

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

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

Step 5.3.1.4.6

Add $1$ and $1$ .

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

Step 5.3.2

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

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

Step 5.4

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

Tap for more steps...

Step 5.4.1

Regroup terms.

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

Step 5.4.2

Rearrange terms.

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

Step 5.4.3

Apply pythagorean identity.

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 5.4.5

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

Tap for more steps...

Step 5.4.5.1

Factor $2$ out of $2$ .

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

Step 5.4.5.2

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

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

Step 5.4.5.3

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

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

Step 6

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

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Separate fractions.

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

Step 8

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

$\frac{2}{1} \sec (x)$

Step 9

Divide $2$ by $1$ .

$2 \sec (x)$