Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

$\csc^{2} (x)$