Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

Separate fractions.

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

Step 7

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

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

Step 8

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

$\sec (x) \tan (x)$