Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Rewrite $\frac{\frac{1}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 1.3

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

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

Step 2

Simplify each term.

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Step 2.1

Separate fractions.

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

Step 2.2

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

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

Step 2.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) \sec (x) - \frac{\sin (x)}{\cos (x)}$

Step 2.4

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

$\csc (x) \sec (x) - \tan (x)$