Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)} - \sin (x)$ .

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Step 1.2.1

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.2.2

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Step 1.2.3

Factor $\sin (x)$ out of $\sin (x) \frac{1}{\cos (x)} + \sin (x) \cdot - 1$ .

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

Step 2

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

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

Step 3

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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Step 5.1

Factor $\sin (x)$ out of $2 \sin (x)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

$(\frac{1}{\cos (x)} - 1) \frac{\cos (x)}{2}$

Step 6

Apply the distributive property.

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{1}{2} - 1 \frac{\cos (x)}{2}$

Step 8

Rewrite $- 1 \frac{\cos (x)}{2}$ as $- \frac{\cos (x)}{2}$ .

$\frac{1}{2} - \frac{\cos (x)}{2}$