Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$\frac{\sec (x)}{\frac{\sin (x)}{\cos (x)}} + \frac{\cot (x)}{\cos (x)}$

Step 1.2

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

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

Step 1.3

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

$\frac{1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cot (x)}{\cos (x)}$

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

$\frac{1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cot (x)}{\cos (x)}$

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

$\frac{1}{\sin (x)} + \frac{\frac{\cos (x)}{\sin (x)}}{\cos (x)}$

Step 1.6

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\cos (x)}$ as a product.

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

Step 1.7

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

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Step 1.7.1

Cancel the common factor.

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

Step 1.7.2

Rewrite the expression.

$\frac{1}{\sin (x)} + \frac{1}{\sin (x)}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 + 1}{\sin (x)}$

Step 2.2

Add $1$ and $1$ .

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

Step 3

Separate fractions.

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

Step 4

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

$\frac{2}{1} \csc (x)$

Step 5

Divide $2$ by $1$ .

$2 \csc (x)$