Algebra Examples

$\frac{\tan (A) (\sec (A) - \cos (A))}{\sin (A)}$

Step 1

Separate fractions.

$\frac{\sec (A) - \cos (A)}{1} \cdot \frac{\tan (A)}{\sin (A)}$

Step 2

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

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

Step 3

Rewrite $\frac{\frac{\sin (A)}{\cos (A)}}{\sin (A)}$ as a product.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{\sec (A) - \cos (A)}{1} \cdot \frac{1}{\cos (A)}$

Step 5

Divide $\sec (A) - \cos (A)$ by $1$ .

$(\sec (A) - \cos (A)) \frac{1}{\cos (A)}$

Step 6

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

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

Step 7

Apply the distributive property.

$\frac{1}{\cos (A)} \cdot \frac{1}{\cos (A)} - \cos (A) \frac{1}{\cos (A)}$

Step 8

Multiply $\frac{1}{\cos (A)} \cdot \frac{1}{\cos (A)}$ .

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Step 8.1

Multiply $\frac{1}{\cos (A)}$ by $\frac{1}{\cos (A)}$ .

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

Step 8.2

Raise $\cos (A)$ to the power of $1$ .

$\frac{1}{\cos^{1} (A) \cos (A)} - \cos (A) \frac{1}{\cos (A)}$

Step 8.3

Raise $\cos (A)$ to the power of $1$ .

$\frac{1}{\cos^{1} (A) \cos^{1} (A)} - \cos (A) \frac{1}{\cos (A)}$

Step 8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.5

Add $1$ and $1$ .

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

Step 9

Simplify terms.

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Step 9.1

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

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Step 9.1.1

Factor $\cos (A)$ out of $- \cos (A)$ .

$\frac{1}{\cos^{2} (A)} + \cos (A) \cdot - 1 \frac{1}{\cos (A)}$

Step 9.1.2

Cancel the common factor.

$\frac{1}{\cos^{2} (A)} + \cos (A) \cdot - 1 \frac{1}{\cos (A)}$

Step 9.1.3

Rewrite the expression.

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

Step 9.2

Simplify each term.

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Step 9.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 9.2.2

Rewrite $\frac{1^{2}}{\cos^{2} (A)}$ as ${(\frac{1}{\cos (A)})}^{2}$ .

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

Step 9.2.3

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

$\sec^{2} (A) - 1$

Step 10

Apply pythagorean identity.

$\tan^{2} (A)$