Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine $- \tan (x)$ and $\frac{1 - \sin (x)}{1 - \sin (x)}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

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

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