Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Divide $\sin (x)$ by $1$ .

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

Step 1.5.2

Combine $\frac{1}{\cos (x)}$ and $\sin (x)$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $\sin (x)$ and $\sin (x)$ .

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

Step 3

Separate fractions.

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

Step 4

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

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

Step 5

Divide $2$ by $1$ .

$2 \tan (x)$