Trigonometry Examples

$\frac{\tan (x)}{\sec^{3} (x)}$

Step 1

Factor $\sec (x)$ out of $\sec^{3} (x)$ .

$\frac{\tan (x)}{\sec (x) \sec^{2} (x)}$

Step 2

Separate fractions.

$\frac{1}{\sec^{2} (x)} \cdot \frac{\tan (x)}{\sec (x)}$

Step 3

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

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

Step 4

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

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

Step 5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 6

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 7

Cancel the common factor of $\cos (x)$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

Step 10

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

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

Step 11

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 12

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

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