Expand Using Sum/Difference Formulas tan(45-30)

$\tan (45 - 30)$

Step 1

Use the difference formula for tangent to simplify the expression. The formula states that $\tan (A - B) = \frac{\tan (A) - \tan (B)}{1 + \tan (A) \tan (B)}$ .

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

Step 2

Remove parentheses.

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

Step 3

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 4

Simplify the denominator.

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

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

Multiply $\tan (30)$ by $1$ .

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

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

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

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 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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 7

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 8

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.

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

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 $\frac{3 - \sqrt{3}}{6}$ and $\sqrt{3}$ .

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

Step 9

Simplify each term.

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

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

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 10

Combine the numerators over the common denominator.

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

Step 11

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Step 12

Simplify terms.

Add $3$ and $1$ .

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

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

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

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 13

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{3}$

Decimal Form:

$0.26794919 \dots$

Expand Using Sum/Difference Formulas tan(45-30)

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