Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Divide $\sec^{2} (x)$ by $1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

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

Step 10

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

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

Step 11

Separate fractions.

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

Step 12

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

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

Step 13

Separate fractions.

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

Step 14

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

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

Step 15

Divide $1^{2}$ by $1$ .

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

Step 16

One to any power is one.

$1 \csc (x) \sec (x) - \tan (x)$

Step 17

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

$\csc (x) \sec (x) - \tan (x)$