Trigonometry Examples

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

Step 1

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

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Step 1.1

Separate fractions.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

$\cot (x) + \tan (x) = \frac{1^{2}}{\cos^{2} (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 1.9

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

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Step 1.9.1

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

$\cot (x) + \tan (x) = \frac{1^{2}}{\cos (x) \cos (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 1.9.2

Cancel the common factor.

$\cot (x) + \tan (x) = \frac{1^{2}}{\cos (x) \cos (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 1.9.3

Rewrite the expression.

$\cot (x) + \tan (x) = \frac{1^{2}}{\cos (x)} \cdot \frac{1}{\sin (x)}$

Step 1.10

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

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

Step 1.11

Separate fractions.

$\cot (x) + \tan (x) = \frac{1^{2}}{\sin (x)} \cdot \frac{1}{\cos (x)}$

Step 1.12

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

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

Step 1.13

Separate fractions.

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

One to any power is one.

$\cot (x) + \tan (x) = 1 \csc (x) \sec (x)$

Step 1.17

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

$\cot (x) + \tan (x) = \csc (x) \sec (x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.1, 0.1, - 0.3, 0.3, - \frac{2}{5}, \frac{2}{5}, \frac{1}{2}$

Step 3