Trigonometry Examples

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

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

Factor $\tan (x)$ out of $\tan (x) + \cot^{2} (x) \tan (x)$ .

Tap for more steps...

Step 1.1.1

Multiply by $1$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor $\tan (x)$ out of $\tan (x) \cdot 1 + \tan (x) \cot^{2} (x)$ .

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

Step 1.2

Apply pythagorean identity.

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

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

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

Step 5

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

$\frac{\sec^{2} (x)}{\csc^{2} (x)} \cot (x)$

Step 6

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

${(\frac{\sec (x)}{\csc (x)})}^{2} \cot (x)$

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\tan^{2} (x) \cot (x)$

Step 11

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

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

Step 12

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

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

Step 13

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 14

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

Tap for more steps...

Step 14.1

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

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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

Step 15

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

Tap for more steps...

Step 15.1

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

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

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

$\tan (x)$