Find the Exact Value tan(105)

$\tan (105)$

Split $105$ into two angles where the values of the six trigonometric functions are known.

$\tan (45 + 60)$

Apply the sum of angles identity.

$\frac{\tan (45) + \tan (60)}{1 - \tan (45) \tan (60)}$

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

$\frac{1 + \tan (60)}{1 - \tan (45) \tan (60)}$

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{1 + \sqrt{3}}{1 - \tan (45) \tan (60)}$

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

$\frac{1 + \sqrt{3}}{1 - 1 \cdot 1 \tan (60)}$

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

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

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Simplify the denominator.

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Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

$\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{9}}{- 2}$

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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Factor $2$ out of $4$ .

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

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

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

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

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

Move the negative one from the denominator of $\frac{2 + \sqrt{3}}{- 1}$ .

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

Rewrite $- 1 \cdot (2 + \sqrt{3})$ as $- (2 + \sqrt{3})$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$

Find the Exact Value tan(105)

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