Precalculus Examples

\frac{sec (x) - cos (x)}{tan (x)}

$\frac{\sec (x) - \cos (x)}{\tan (x)}$

Step 1

Rewrite

sec (x)

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

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

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

Step 2

Rewrite

tan (x)

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

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 4

Apply the distributive property.

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

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

Step 5

Cancel the common factor of

cos (x)

$\cos (x)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Multiply

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

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Step 6.1

Combine

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

$\cos (x)$ .

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

Step 6.2

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 6.3

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 6.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5

Add

$1$ and

$1$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Apply pythagorean identity.

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

Step 9

Cancel the common factor of

$\sin^{2} (x)$ and

$\sin (x)$ .

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Step 9.1

Factor

$\sin (x)$ out of

$\sin^{2} (x)$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Multiply by

$1$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 9.2.4

Divide

$\sin (x)$ by

$1$ .

$\sin (x)$

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