Trigonometry Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Rewrite $\frac{\frac{1}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Rewrite $\frac{\frac{1}{\sin (x)}}{\cos (x)}$ as a product.

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

Step 1.6

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

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

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $1$ and $1$ .

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

Step 3

Separate fractions.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

Divide $2$ by $1$ .

$2 \csc (x) \sec (x)$