Algebra Examples

\frac{csc (x) cos (x)}{tan (x) + cot (x)}

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

Step 1

Rewrite

csc (x)

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

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

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

tan (x)

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

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

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

Step 2.2

Rewrite

cot (x)

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

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

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

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

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

Step 3

Combine

\frac{1}{sin (x)}

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

cos (x)

$\cos (x)$ .

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 5

Multiply

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

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

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

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

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

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

Step 6

Separate fractions.

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

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

Step 7

Convert from

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

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

cot (x)

$\cot (x)$ .

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

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

Step 8

Simplify each term.

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Step 8.1

Convert from

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

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

tan (x)

$\tan (x)$ .

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

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

Step 8.2

Convert from

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

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

cot (x)

$\cot (x)$ .

\frac{1}{tan (x) + cot (x)} cot (x)

$\frac{1}{\tan (x) + \cot (x)} \cot (x)$

\frac{1}{tan (x) + cot (x)} cot (x)

$\frac{1}{\tan (x) + \cot (x)} \cot (x)$

Step 9

Combine

\frac{1}{tan (x) + cot (x)}

$\frac{1}{\tan (x) + \cot (x)}$ and

cot (x)

$\cot (x)$ .

\frac{cot (x)}{tan (x) + cot (x)}

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