Trigonometry Examples

$\frac{\tan (\frac{5 π}{12}) + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1

Simplify the numerator.

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Step 1.1

The exact value of $\tan (\frac{5 π}{12})$ is $2 + \sqrt{3}$ .

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Step 1.1.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{\tan (\frac{π}{6} + \frac{π}{4}) + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.2

Apply the sum of angles identity.

$\frac{\frac{\tan (\frac{π}{6}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{4})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 1.1.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.1.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

$\frac{\frac{3}{3} \cdot \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.1.2

Combine.

$\frac{\frac{3 (\frac{\sqrt{3}}{3} + 1)}{3 (1 - \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.2

Apply the distributive property.

$\frac{\frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.3

Cancel the common factor of $3$ .

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Step 1.1.7.3.1

Cancel the common factor.

$\frac{\frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.3.2

Rewrite the expression.

$\frac{\frac{\sqrt{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.4

Multiply $3$ by $1$ .

$\frac{\frac{\sqrt{3} + 3}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.5

Simplify the denominator.

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Step 1.1.7.5.1

Multiply $3$ by $1$ .

$\frac{\frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.5.2

Multiply $- 1$ by $1$ .

$\frac{\frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.5.3

Cancel the common factor of $3$ .

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Step 1.1.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{\frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.5.3.2

Cancel the common factor.

$\frac{\frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.5.3.3

Rewrite the expression.

$\frac{\frac{\sqrt{3} + 3}{3 - \sqrt{3}} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.6

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{\frac{\sqrt{3} + 3}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.7

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{\frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.8

Expand the denominator using the FOIL method.

$\frac{\frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.9

Simplify.

$\frac{\frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.10

Simplify the numerator.

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Step 1.1.7.10.1

Reorder terms.

$\frac{\frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.10.2

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{\frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.10.3

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{\frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.10.4

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

$\frac{\frac{{(3 + \sqrt{3})}^{1 + 1}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.10.5

Add $1$ and $1$ .

$\frac{\frac{{(3 + \sqrt{3})}^{2}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{\frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.12

Expand $(3 + \sqrt{3}) (3 + \sqrt{3})$ using the FOIL Method.

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Step 1.1.7.12.1

Apply the distributive property.

$\frac{\frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.12.2

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.12.3

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13

Simplify and combine like terms.

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Step 1.1.7.13.1

Simplify each term.

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Step 1.1.7.13.1.1

Multiply $3$ by $3$ .

$\frac{\frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.1.2

Move $3$ to the left of $\sqrt{3}$ .

$\frac{\frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.1.3

Combine using the product rule for radicals.

$\frac{\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.1.4

Multiply $3$ by $3$ .

$\frac{\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.2

Add $9$ and $3$ .

$\frac{\frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{\frac{12 + 6 \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 1.1.7.14.1

Factor $6$ out of $12$ .

$\frac{\frac{6 \cdot 2 + 6 \sqrt{3}}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{\frac{6 \cdot 2 + 6 (\sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{\frac{6 (2 + \sqrt{3})}{6} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.4

Cancel the common factors.

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Step 1.1.7.14.4.1

Factor $6$ out of $6$ .

$\frac{\frac{6 (2 + \sqrt{3})}{6 (1)} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.4.2

Cancel the common factor.

$\frac{\frac{6 (2 + \sqrt{3})}{6 \cdot 1} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.4.3

Rewrite the expression.

$\frac{\frac{2 + \sqrt{3}}{1} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.1.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$\frac{2 + \sqrt{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 + \sqrt{3} + 1}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 1.3

Add $2$ and $1$ .

$\frac{3 + \sqrt{3}}{1 - \tan (\frac{5 π}{12}) \tan (\frac{π}{4})}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\tan (\frac{5 π}{12})$ is $2 + \sqrt{3}$ .

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Step 2.1.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{3 + \sqrt{3}}{1 - \tan (\frac{π}{6} + \frac{π}{4}) \tan (\frac{π}{4})}$

Step 2.1.2

Apply the sum of angles identity.

$\frac{3 + \sqrt{3}}{1 - \frac{\tan (\frac{π}{6}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} \tan (\frac{π}{4})}$

Step 2.1.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} \tan (\frac{π}{4})}$

Step 2.1.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})} \tan (\frac{π}{4})}$

Step 2.1.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{4})} \tan (\frac{π}{4})}$

Step 2.1.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1} \tan (\frac{π}{4})}$

Step 2.1.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 2.1.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.1.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

$\frac{3 + \sqrt{3}}{1 - (\frac{3}{3} \cdot \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}) \tan (\frac{π}{4})}$

Step 2.1.7.1.2

Combine.

$\frac{3 + \sqrt{3}}{1 - \frac{3 (\frac{\sqrt{3}}{3} + 1)}{3 (1 - \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.2

Apply the distributive property.

$\frac{3 + \sqrt{3}}{1 - \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.3

Cancel the common factor of $3$ .

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Step 2.1.7.3.1

Cancel the common factor.

$\frac{3 + \sqrt{3}}{1 - \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.3.2

Rewrite the expression.

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.4

Multiply $3$ by $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.5

Simplify the denominator.

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Step 2.1.7.5.1

Multiply $3$ by $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)} \tan (\frac{π}{4})}$

Step 2.1.7.5.2

Multiply $- 1$ by $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})} \tan (\frac{π}{4})}$

Step 2.1.7.5.3

Cancel the common factor of $3$ .

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Step 2.1.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}} \tan (\frac{π}{4})}$

Step 2.1.7.5.3.2

Cancel the common factor.

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}} \tan (\frac{π}{4})}$

Step 2.1.7.5.3.3

Rewrite the expression.

$\frac{3 + \sqrt{3}}{1 - \frac{\sqrt{3} + 3}{3 - \sqrt{3}} \tan (\frac{π}{4})}$

Step 2.1.7.6

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{3 + \sqrt{3}}{1 - (\frac{\sqrt{3} + 3}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}}) \tan (\frac{π}{4})}$

Step 2.1.7.7

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})} \tan (\frac{π}{4})}$

Step 2.1.7.8

Expand the denominator using the FOIL method.

$\frac{3 + \sqrt{3}}{1 - \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}} \tan (\frac{π}{4})}$

Step 2.1.7.9

Simplify.

$\frac{3 + \sqrt{3}}{1 - \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.10

Simplify the numerator.

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Step 2.1.7.10.1

Reorder terms.

$\frac{3 + \sqrt{3}}{1 - \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.10.2

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.10.3

Raise $3 + \sqrt{3}$ to the power of $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.10.4

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

$\frac{3 + \sqrt{3}}{1 - \frac{{(3 + \sqrt{3})}^{1 + 1}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.10.5

Add $1$ and $1$ .

$\frac{3 + \sqrt{3}}{1 - \frac{{(3 + \sqrt{3})}^{2}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{3 + \sqrt{3}}{1 - \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.12

Expand $(3 + \sqrt{3}) (3 + \sqrt{3})$ using the FOIL Method.

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Step 2.1.7.12.1

Apply the distributive property.

$\frac{3 + \sqrt{3}}{1 - \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.12.2

Apply the distributive property.

$\frac{3 + \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.12.3

Apply the distributive property.

$\frac{3 + \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13

Simplify and combine like terms.

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Step 2.1.7.13.1

Simplify each term.

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Step 2.1.7.13.1.1

Multiply $3$ by $3$ .

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.1.2

Move $3$ to the left of $\sqrt{3}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.1.3

Combine using the product rule for radicals.

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.1.4

Multiply $3$ by $3$ .

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 + \sqrt{3}}{1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.2

Add $9$ and $3$ .

$\frac{3 + \sqrt{3}}{1 - \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{12 + 6 \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 2.1.7.14.1

Factor $6$ out of $12$ .

$\frac{3 + \sqrt{3}}{1 - \frac{6 \cdot 2 + 6 \sqrt{3}}{6} \tan (\frac{π}{4})}$

Step 2.1.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{3 + \sqrt{3}}{1 - \frac{6 \cdot 2 + 6 (\sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{3 + \sqrt{3}}{1 - \frac{6 (2 + \sqrt{3})}{6} \tan (\frac{π}{4})}$

Step 2.1.7.14.4

Cancel the common factors.

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Step 2.1.7.14.4.1

Factor $6$ out of $6$ .

$\frac{3 + \sqrt{3}}{1 - \frac{6 (2 + \sqrt{3})}{6 (1)} \tan (\frac{π}{4})}$

Step 2.1.7.14.4.2

Cancel the common factor.

$\frac{3 + \sqrt{3}}{1 - \frac{6 (2 + \sqrt{3})}{6 \cdot 1} \tan (\frac{π}{4})}$

Step 2.1.7.14.4.3

Rewrite the expression.

$\frac{3 + \sqrt{3}}{1 - \frac{2 + \sqrt{3}}{1} \tan (\frac{π}{4})}$

Step 2.1.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$\frac{3 + \sqrt{3}}{1 - (2 + \sqrt{3}) \tan (\frac{π}{4})}$

Step 2.2

Apply the distributive property.

$\frac{3 + \sqrt{3}}{1 + (- 1 \cdot 2 - \sqrt{3}) \tan (\frac{π}{4})}$

Step 2.3

Multiply $- 1$ by $2$ .

$\frac{3 + \sqrt{3}}{1 + (- 2 - \sqrt{3}) \tan (\frac{π}{4})}$

Step 2.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{3 + \sqrt{3}}{1 + (- 2 - \sqrt{3}) \cdot 1}$

Step 2.5

Multiply $- 2 - \sqrt{3}$ by $1$ .

$\frac{3 + \sqrt{3}}{1 - 2 - \sqrt{3}}$

Step 2.6

Subtract $2$ from $1$ .

$\frac{3 + \sqrt{3}}{- 1 - \sqrt{3}}$

Step 3

Multiply $\frac{3 + \sqrt{3}}{- 1 - \sqrt{3}}$ by $\frac{- 1 + \sqrt{3}}{- 1 + \sqrt{3}}$ .

$\frac{3 + \sqrt{3}}{- 1 - \sqrt{3}} \cdot \frac{- 1 + \sqrt{3}}{- 1 + \sqrt{3}}$

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{3 + \sqrt{3}}{- 1 - \sqrt{3}}$ by $\frac{- 1 + \sqrt{3}}{- 1 + \sqrt{3}}$ .

$\frac{(3 + \sqrt{3}) (- 1 + \sqrt{3})}{(- 1 - \sqrt{3}) (- 1 + \sqrt{3})}$

Step 4.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{3}) (- 1 + \sqrt{3})}{1 - \sqrt{3} + \sqrt{3} - {\sqrt{3}}^{2}}$

Step 4.3

Simplify.

$\frac{(3 + \sqrt{3}) (- 1 + \sqrt{3})}{- 2}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(3 + \sqrt{3}) (- 1 + \sqrt{3})$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{3 (- 1 + \sqrt{3}) + \sqrt{3} (- 1 + \sqrt{3})}{- 2}$

Step 5.1.2

Apply the distributive property.

$\frac{3 \cdot - 1 + 3 \sqrt{3} + \sqrt{3} (- 1 + \sqrt{3})}{- 2}$

Step 5.1.3

Apply the distributive property.

$\frac{3 \cdot - 1 + 3 \sqrt{3} + \sqrt{3} \cdot - 1 + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $3$ by $- 1$ .

$\frac{- 3 + 3 \sqrt{3} + \sqrt{3} \cdot - 1 + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.2.1.2

Move $- 1$ to the left of $\sqrt{3}$ .

$\frac{- 3 + 3 \sqrt{3} - 1 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.2.1.3

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\frac{- 3 + 3 \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}}{- 2}$

Step 5.2.1.4

Combine using the product rule for radicals.

$\frac{- 3 + 3 \sqrt{3} - \sqrt{3} + \sqrt{3 \cdot 3}}{- 2}$

Step 5.2.1.5

Multiply $3$ by $3$ .

$\frac{- 3 + 3 \sqrt{3} - \sqrt{3} + \sqrt{9}}{- 2}$

Step 5.2.1.6

Rewrite $9$ as $3^{2}$ .

$\frac{- 3 + 3 \sqrt{3} - \sqrt{3} + \sqrt{3^{2}}}{- 2}$

Step 5.2.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- 3 + 3 \sqrt{3} - \sqrt{3} + 3}{- 2}$

Step 5.2.2

Add $- 3$ and $3$ .

$\frac{0 + 3 \sqrt{3} - \sqrt{3}}{- 2}$

Step 5.2.3

Add $0$ and $3 \sqrt{3}$ .

$\frac{3 \sqrt{3} - \sqrt{3}}{- 2}$

Step 5.2.4

Subtract $\sqrt{3}$ from $3 \sqrt{3}$ .

$\frac{2 \sqrt{3}}{- 2}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $2$ and $- 2$ .

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Step 6.1.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3})}{- 2}$

Step 6.1.2

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$- 1 \cdot \sqrt{3}$

Step 6.2

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$- \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$