Precalculus Examples

$\frac{\sec (θ) \tan (θ)}{\sin (θ)}$

Step 1

Separate fractions.

$\frac{\tan (θ)}{1} \cdot \frac{\sec (θ)}{\sin (θ)}$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Divide $\tan (θ)$ by $1$ .

$\tan (θ) \frac{1}{\cos (θ) \sin (θ)}$

Step 6

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

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

Step 7

Cancel the common factor of $\sin (θ)$ .

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Step 7.1

Factor $\sin (θ)$ out of $\cos (θ) \sin (θ)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{1}{\cos (θ)} \cdot \frac{1}{\cos (θ)}$

Step 8

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

$\frac{1}{\cos (θ) \cos (θ)}$

Step 9

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

$\frac{1}{\cos^{1} (θ) \cos (θ)}$

Step 10

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

$\frac{1}{\cos^{1} (θ) \cos^{1} (θ)}$

Step 11

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

$\frac{1}{{\cos (θ)}^{1 + 1}}$

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $1$ .

$\frac{1}{\cos^{2} (θ)}$

Step 12.2

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\cos^{2} (θ)}$

Step 13

Rewrite $\frac{1^{2}}{\cos^{2} (θ)}$ as ${(\frac{1}{\cos (θ)})}^{2}$ .

${(\frac{1}{\cos (θ)})}^{2}$

Step 14

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

$\sec^{2} (θ)$