Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 2

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

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $\frac{1}{\cos (x)} - 1$ by $1$ .

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

Step 3

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

$\sec (x) - 1$