Algebra Examples

$f (x) = (\frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 1

Write $f (x) = (\frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$ as an equation.

$y = (\frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2

Simplify $(\frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$ .

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Step 2.1

To write $\frac{1}{\sec (x) - 1}$ as a fraction with a common denominator, multiply by $\frac{\sec (x) + 1}{\sec (x) + 1}$ .

$y = (\frac{1}{\sec (x) - 1} \cdot \frac{\sec (x) + 1}{\sec (x) + 1} + \frac{1}{\sec (x) + 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.2

To write $\frac{1}{\sec (x) + 1}$ as a fraction with a common denominator, multiply by $\frac{\sec (x) - 1}{\sec (x) - 1}$ .

$y = (\frac{1}{\sec (x) - 1} \cdot \frac{\sec (x) + 1}{\sec (x) + 1} + \frac{1}{\sec (x) + 1} \cdot \frac{\sec (x) - 1}{\sec (x) - 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.3

Write each expression with a common denominator of $(\sec (x) - 1) (\sec (x) + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{\sec (x) - 1}$ by $\frac{\sec (x) + 1}{\sec (x) + 1}$ .

$y = (\frac{\sec (x) + 1}{(\sec (x) - 1) (\sec (x) + 1)} + \frac{1}{\sec (x) + 1} \cdot \frac{\sec (x) - 1}{\sec (x) - 1}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.3.2

Multiply $\frac{1}{\sec (x) + 1}$ by $\frac{\sec (x) - 1}{\sec (x) - 1}$ .

$y = (\frac{\sec (x) + 1}{(\sec (x) - 1) (\sec (x) + 1)} + \frac{\sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.3.3

Reorder the factors of $(\sec (x) - 1) (\sec (x) + 1)$ .

$y = (\frac{\sec (x) + 1}{(\sec (x) + 1) (\sec (x) - 1)} + \frac{\sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)}) \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.4

Combine the numerators over the common denominator.

$y = \frac{\sec (x) + 1 + \sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Add $\sec (x)$ and $\sec (x)$ .

$y = \frac{1 + 2 \sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.5.2

Subtract $1$ from $1$ .

$y = \frac{0 + 2 \sec (x)}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.5.3

Add $0$ and $2 \sec (x)$ .

$y = \frac{2 \sec (x)}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\tan^{2} (x)}{\csc (x)}$

Step 2.6

Combine.

$y = \frac{2 \sec (x) \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1) \csc (x)}$

Step 2.7

Separate fractions.

$y = \frac{2 \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\sec (x)}{\csc (x)}$

Step 2.8

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

$y = \frac{2 \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1)} \cdot \frac{\sec (x)}{\frac{1}{\sin (x)}}$

Step 2.9

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

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

Step 2.10

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 2.11

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

$y = \frac{2 \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1)} \tan (x)$

Step 2.12

Multiply $\frac{2 \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1)} \tan (x)$ .

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Step 2.12.1

Combine $\frac{2 \tan^{2} (x)}{(\sec (x) + 1) (\sec (x) - 1)}$ and $\tan (x)$ .

$y = \frac{2 \tan^{2} (x) \tan (x)}{(\sec (x) + 1) (\sec (x) - 1)}$

Step 2.12.2

Raise $\tan (x)$ to the power of $1$ .

$y = \frac{2 (\tan^{1} (x) \tan^{2} (x))}{(\sec (x) + 1) (\sec (x) - 1)}$

Step 2.12.3

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

$y = \frac{2 {\tan (x)}^{1 + 2}}{(\sec (x) + 1) (\sec (x) - 1)}$

Step 2.12.4

Add $1$ and $2$ .

$y = \frac{2 \tan^{3} (x)}{(\sec (x) + 1) (\sec (x) - 1)}$