Find the Exact Value (tan(45 degrees )+tan(30 degrees ))/(1-tan(45 degrees )tan(30 degrees ))

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

Step 1

Simplify the numerator.

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

Step 6

Simplify terms.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

$(3 + \sqrt{3}) \frac{3 + \sqrt{3}}{6}$

Apply the distributive property.

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\sqrt{3}$ and $\frac{3 + \sqrt{3}}{6}$ .

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

Step 7

Simplify each term.

Apply the distributive property.

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

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

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

Combine using the product rule for radicals.

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

Simplify each term.

Multiply $3$ by $3$ .

$\frac{3 + \sqrt{3}}{2} + \frac{3 \sqrt{3} + \sqrt{9}}{6}$

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

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

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

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

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

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

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $3 (\sqrt{3}) + 3 (1)$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Simplify terms.

Combine the numerators over the common denominator.

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

Add $3$ and $1$ .

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

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

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

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

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$2 + \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$

Find the Exact Value (tan(45 degrees )+tan(30 degrees ))/(1-tan(45 degrees )tan(30 degrees ))

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