Algebra Examples

$\frac{\sec (θ) \sin (θ)}{\tan (θ) + \cot (θ)}$

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Separate fractions.

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Combine $\frac{1}{\tan (θ) + \cot (θ)}$ and $\tan (θ)$ .

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