Find the Exact Value (tan(15 degrees )+tan(15 degrees ))/(1-tan(15 degrees )tan(15 degrees ))

$\frac{\tan (15 °) + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Step 1

Simplify the numerator.

The exact value of $\tan (15 °)$ is $2 - \sqrt{3}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\tan (45 - 30) + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Separate negation.

$\frac{\tan (45 - (30)) + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Apply the difference of angles identity.

$\frac{\frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{\frac{1 - \tan (30)}{1 + \tan (45) \tan (30)} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

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

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

$\frac{\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Combine.

$\frac{\frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{\frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $3$ .

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

$\frac{\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{\frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{\frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Multiply $3$ by $1$ .

$\frac{\frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Simplify the denominator.

Multiply $3$ by $1$ .

$\frac{\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 \cdot 1$ .

$\frac{\frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{\frac{3 - \sqrt{3}}{3 + \sqrt{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Expand the denominator using the FOIL method.

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Simplify.

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Simplify the numerator.

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

$\frac{\frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{{(3 - \sqrt{3})}^{1 + 1}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Add $1$ and $1$ .

$\frac{\frac{{(3 - \sqrt{3})}^{2}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

Apply the distributive property.

$\frac{\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$\frac{\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Multiply $- 1$ by $3$ .

$\frac{\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Multiply $3$ by $- 1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Add $1$ and $1$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Evaluate the exponent.

$\frac{\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Add $9$ and $3$ .

$\frac{\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{12 - 6 \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

Factor $6$ out of $12$ .

$\frac{\frac{6 \cdot 2 - 6 \sqrt{3}}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{6 \cdot 2 + 6 (- \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{\frac{6 (2 - \sqrt{3})}{6} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factors.

Factor $6$ out of $6$ .

$\frac{\frac{6 (2 - \sqrt{3})}{6 (1)} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{\frac{6 (2 - \sqrt{3})}{6 \cdot 1} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{\frac{2 - \sqrt{3}}{1} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \tan (15 °)}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (15 °)$ is $2 - \sqrt{3}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{2 - \sqrt{3} + \tan (45 - 30)}{1 - \tan (15 °) \tan (15 °)}$

Separate negation.

$\frac{2 - \sqrt{3} + \tan (45 - (30))}{1 - \tan (15 °) \tan (15 °)}$

Apply the difference of angles identity.

$\frac{2 - \sqrt{3} + \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{2 - \sqrt{3} + \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}}{1 - \tan (15 °) \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 - \sqrt{3} + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

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

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

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

$\frac{2 - \sqrt{3} + \frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

Combine.

$\frac{2 - \sqrt{3} + \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})}}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $3$ .

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

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}}{1 - \tan (15 °) \tan (15 °)}$

Multiply $3$ by $1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

Simplify the denominator.

Multiply $3$ by $1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 \cdot 1$ .

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + \sqrt{3}}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}}{1 - \tan (15 °) \tan (15 °)}$

Expand the denominator using the FOIL method.

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}}{1 - \tan (15 °) \tan (15 °)}$

Simplify.

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Simplify the numerator.

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Add $1$ and $1$ .

$\frac{2 - \sqrt{3} + \frac{{(3 - \sqrt{3})}^{2}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Apply the distributive property.

$\frac{2 - \sqrt{3} + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$\frac{2 - \sqrt{3} + \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Multiply $- 1$ by $3$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Multiply $3$ by $- 1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Add $1$ and $1$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}}{1 - \tan (15 °) \tan (15 °)}$

Evaluate the exponent.

$\frac{2 - \sqrt{3} + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}}{1 - \tan (15 °) \tan (15 °)}$

Add $9$ and $3$ .

$\frac{2 - \sqrt{3} + \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{12 - 6 \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

Factor $6$ out of $12$ .

$\frac{2 - \sqrt{3} + \frac{6 \cdot 2 - 6 \sqrt{3}}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factors.

Factor $6$ out of $6$ .

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6 (1)}}{1 - \tan (15 °) \tan (15 °)}$

Cancel the common factor.

$\frac{2 - \sqrt{3} + \frac{6 (2 - \sqrt{3})}{6 \cdot 1}}{1 - \tan (15 °) \tan (15 °)}$

Rewrite the expression.

$\frac{2 - \sqrt{3} + \frac{2 - \sqrt{3}}{1}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{2 - \sqrt{3} + 2 - \sqrt{3}}{1 - \tan (15 °) \tan (15 °)}$

Add $2$ and $2$ .

$\frac{4 - \sqrt{3} - \sqrt{3}}{1 - \tan (15 °) \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \tan (15 °) \tan (15 °)}$

Step 2

Simplify the denominator.

The exact value of $\tan (15 °)$ is $2 - \sqrt{3}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{4 - 2 \sqrt{3}}{1 - \tan (45 - 30) \tan (15 °)}$

Separate negation.

$\frac{4 - 2 \sqrt{3}}{1 - \tan (45 - (30)) \tan (15 °)}$

Apply the difference of angles identity.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)} \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)} \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)} \tan (15 °)}$

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)} \tan (15 °)}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}} \tan (15 °)}$

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

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

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

$\frac{4 - 2 \sqrt{3}}{1 - (\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}) \tan (15 °)}$

Combine.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})} \tan (15 °)}$

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15 °)}$

Cancel the common factor of $3$ .

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15 °)}$

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15 °)}$

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})} \tan (15 °)}$

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15 °)}$

Simplify the denominator.

Multiply $3$ by $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15 °)}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 \cdot 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}} \tan (15 °)}$

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}} \tan (15 °)}$

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - (\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}) \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})} \tan (15 °)}$

Expand the denominator using the FOIL method.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}} \tan (15 °)}$

Simplify.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \tan (15 °)}$

Simplify the numerator.

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{1 + 1}}{6} \tan (15 °)}$

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{{(3 - \sqrt{3})}^{2}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6} \tan (15 °)}$

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

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \tan (15 °)}$

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6} \tan (15 °)}$

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15 °)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15 °)}$

Multiply $- 1$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6} \tan (15 °)}$

Multiply $3$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6} \tan (15 °)}$

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6} \tan (15 °)}$

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6} \tan (15 °)}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6} \tan (15 °)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6} \tan (15 °)}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \tan (15 °)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6} \tan (15 °)}$

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6} \tan (15 °)}$

Evaluate the exponent.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6} \tan (15 °)}$

Add $9$ and $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{12 - 6 \sqrt{3}}{6} \tan (15 °)}$

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

Factor $6$ out of $12$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 \cdot 2 - 6 \sqrt{3}}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6} \tan (15 °)}$

Cancel the common factors.

Factor $6$ out of $6$ .

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6 (1)} \tan (15 °)}$

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{6 (2 - \sqrt{3})}{6 \cdot 1} \tan (15 °)}$

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - \frac{2 - \sqrt{3}}{1} \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 - (2 - \sqrt{3}) \tan (15 °)}$

Apply the distributive property.

$\frac{4 - 2 \sqrt{3}}{1 + (- 1 \cdot 2 - - \sqrt{3}) \tan (15 °)}$

Multiply $- 1$ by $2$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 - - \sqrt{3}) \tan (15 °)}$

Multiply $- - \sqrt{3}$ .

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + 1 \sqrt{3}) \tan (15 °)}$

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

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (15 °)}$

The exact value of $\tan (15 °)$ is $2 - \sqrt{3}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (45 - 30)}$

Separate negation.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \tan (45 - (30))}$

Apply the difference of angles identity.

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}}$

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}}$

The exact value of $\tan (45)$ is $1$ .

$\frac{4 - 2 \sqrt{3}}{1 + (- 2 + \sqrt{3}) \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}}$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

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

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

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

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $1$ .

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

Simplify the denominator.

Multiply $3$ by $1$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

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

Multiply $- 1$ by $3$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $9$ and $3$ .

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

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

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

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

Factor $6$ out of $12$ .

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

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

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

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

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

Cancel the common factors.

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Expand $(- 2 + \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $2$ .

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

Multiply $- 1$ by $- 2$ .

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

Move $2$ to the left of $\sqrt{3}$ .

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

Multiply $\sqrt{3} (- \sqrt{3})$ .

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

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

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

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

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

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3^{1}}$

Evaluate the exponent.

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 1 \cdot 3}$

Multiply $- 1$ by $3$ .

$\frac{4 - 2 \sqrt{3}}{1 - 4 + 2 \sqrt{3} + 2 \sqrt{3} - 3}$

Subtract $3$ from $- 4$ .

$\frac{4 - 2 \sqrt{3}}{1 - 7 + 2 \sqrt{3} + 2 \sqrt{3}}$

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{4 - 2 \sqrt{3}}{1 - 7 + 4 \sqrt{3}}$

Subtract $7$ from $1$ .

$\frac{4 - 2 \sqrt{3}}{- 6 + 4 \sqrt{3}}$

Step 3

Cancel the common factor of $4 - 2 \sqrt{3}$ and $- 6 + 4 \sqrt{3}$ .

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $- 6$ .

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

Factor $2$ out of $4 \sqrt{3}$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Multiply $\frac{2 - \sqrt{3}}{- 3 + 2 \sqrt{3}}$ by $\frac{- 3 - 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ .

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

Step 5

Combine fractions.

Multiply $\frac{2 - \sqrt{3}}{- 3 + 2 \sqrt{3}}$ by $\frac{- 3 - 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ .

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 6

Simplify the numerator.

Expand $(2 - \sqrt{3}) (- 3 - 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 3$ .

$\frac{- 6 + 2 (- 2 \sqrt{3}) - \sqrt{3} \cdot - 3 - \sqrt{3} (- 2 \sqrt{3})}{- 3}$

Multiply $- 2$ by $2$ .

$\frac{- 6 - 4 \sqrt{3} - \sqrt{3} \cdot - 3 - \sqrt{3} (- 2 \sqrt{3})}{- 3}$

Multiply $- 3$ by $- 1$ .

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

Multiply $- \sqrt{3} (- 2 \sqrt{3})$ .

Multiply $- 2$ by $- 1$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \sqrt{3} \sqrt{3}}{- 3}$

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

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

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

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

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

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 {\sqrt{3}}^{1 + 1}}{- 3}$

Add $1$ and $1$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 {\sqrt{3}}^{2}}{- 3}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2}}{- 3}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Rewrite the expression.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3^{1}}{- 3}$

Evaluate the exponent.

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 2 \cdot 3}{- 3}$

Multiply $2$ by $3$ .

$\frac{- 6 - 4 \sqrt{3} + 3 \sqrt{3} + 6}{- 3}$

Add $- 6$ and $6$ .

$\frac{0 - 4 \sqrt{3} + 3 \sqrt{3}}{- 3}$

Subtract $4 \sqrt{3}$ from $0$ .

$\frac{- 4 \sqrt{3} + 3 \sqrt{3}}{- 3}$

Add $- 4 \sqrt{3}$ and $3 \sqrt{3}$ .

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

Step 7

Dividing two negative values results in a positive value.

$\frac{\sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$

Find the Exact Value (tan(15 degrees )+tan(15 degrees ))/(1-tan(15 degrees )tan(15 degrees ))

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