Algebra Examples

\frac{1 + \frac{tan (θ) + 1}{tan (θ)}}{- 1 + \frac{tan (θ) - 1}{tan (θ)}}

$\frac{1 + \frac{\tan (θ) + 1}{\tan (θ)}}{- 1 + \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 1

Multiply the numerator and denominator of the fraction by

tan (θ)

$\tan (θ)$ .

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Step 1.1

Multiply

\frac{1 + \frac{tan (θ) + 1}{tan (θ)}}{- 1 + \frac{tan (θ) - 1}{tan (θ)}}

$\frac{1 + \frac{\tan (θ) + 1}{\tan (θ)}}{- 1 + \frac{\tan (θ) - 1}{\tan (θ)}}$ by

\frac{tan (θ)}{tan (θ)}

$\frac{\tan (θ)}{\tan (θ)}$ .

\frac{tan (θ)}{tan (θ)} \cdot \frac{1 + \frac{tan (θ) + 1}{tan (θ)}}{- 1 + \frac{tan (θ) - 1}{tan (θ)}}

$\frac{\tan (θ)}{\tan (θ)} \cdot \frac{1 + \frac{\tan (θ) + 1}{\tan (θ)}}{- 1 + \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 1.2

Combine.

\frac{tan (θ) (1 + \frac{tan (θ) + 1}{tan (θ)})}{tan (θ) (- 1 + \frac{tan (θ) - 1}{tan (θ)})}

$\frac{\tan (θ) (1 + \frac{\tan (θ) + 1}{\tan (θ)})}{\tan (θ) (- 1 + \frac{\tan (θ) - 1}{\tan (θ)})}$

\frac{tan (θ) (1 + \frac{tan (θ) + 1}{tan (θ)})}{tan (θ) (- 1 + \frac{tan (θ) - 1}{tan (θ)})}

$\frac{\tan (θ) (1 + \frac{\tan (θ) + 1}{\tan (θ)})}{\tan (θ) (- 1 + \frac{\tan (θ) - 1}{\tan (θ)})}$

Step 2

Apply the distributive property.

\frac{tan (θ) \cdot 1 + tan (θ) \frac{tan (θ) + 1}{tan (θ)}}{tan (θ) \cdot - 1 + tan (θ) \frac{tan (θ) - 1}{tan (θ)}}

$\frac{\tan (θ) \cdot 1 + \tan (θ) \frac{\tan (θ) + 1}{\tan (θ)}}{\tan (θ) \cdot - 1 + \tan (θ) \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of

tan (θ)

$\tan (θ)$ .

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Step 3.1.1

Cancel the common factor.

$\frac{\tan (θ) \cdot 1 + \tan (θ) \frac{\tan (θ) + 1}{\tan (θ)}}{\tan (θ) \cdot - 1 + \tan (θ) \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 3.1.2

Rewrite the expression.

$\frac{\tan (θ) \cdot 1 + \tan (θ) + 1}{\tan (θ) \cdot - 1 + \tan (θ) \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 3.2

Cancel the common factor of

$\tan (θ)$ .

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Step 3.2.1

Cancel the common factor.

$\frac{\tan (θ) \cdot 1 + \tan (θ) + 1}{\tan (θ) \cdot - 1 + \tan (θ) \frac{\tan (θ) - 1}{\tan (θ)}}$

Step 3.2.2

Rewrite the expression.

$\frac{\tan (θ) \cdot 1 + \tan (θ) + 1}{\tan (θ) \cdot - 1 + \tan (θ) - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply

$\tan (θ)$ by

$1$ .

$\frac{\tan (θ) + \tan (θ) + 1}{\tan (θ) \cdot - 1 + \tan (θ) - 1}$

Step 4.2

Add

$\tan (θ)$ and

$\tan (θ)$ .

$\frac{2 \tan (θ) + 1}{\tan (θ) \cdot - 1 + \tan (θ) - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Move

$- 1$ to the left of

$\tan (θ)$ .

$\frac{2 \tan (θ) + 1}{- 1 \cdot \tan (θ) + \tan (θ) - 1}$

Step 5.2

Rewrite

$- 1 \tan (θ)$ as

$- \tan (θ)$ .

$\frac{2 \tan (θ) + 1}{- \tan (θ) + \tan (θ) - 1}$

Step 5.3

Add

$- \tan (θ)$ and

$\tan (θ)$ .

$\frac{2 \tan (θ) + 1}{0 - 1}$

Step 5.4

Subtract

$1$ from

$0$ .

$\frac{2 \tan (θ) + 1}{- 1}$

Step 6

Simplify by multiplying through.

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Step 6.1

Move the negative one from the denominator of

$\frac{2 \tan (θ) + 1}{- 1}$ .

$- 1 \cdot (2 \tan (θ) + 1)$

Step 6.2

Apply the distributive property.

$- 1 (2 \tan (θ)) - 1 \cdot 1$

Step 6.3

Multiply.

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Step 6.3.1

Multiply

$2$ by

$- 1$ .

$- 2 \tan (θ) - 1 \cdot 1$

Step 6.3.2

Multiply

$- 1$ by

$1$ .

$- 2 \tan (θ) - 1$