Expand Using Sum/Difference Formulas tan(165)

$\tan (165)$

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $165$ can be split into $120 + 45$ .

$\tan (120 + 45)$

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

$\frac{\tan (120) + \tan (45)}{1 - \tan (120) \tan (45)}$

Simplify the numerator.

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Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \tan (60) + \tan (45)}{1 - \tan (120) \tan (45)}$

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

$\frac{- \sqrt{3} + \tan (45)}{1 - \tan (120) \tan (45)}$

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

$\frac{- \sqrt{3} + 1}{1 - \tan (120) \tan (45)}$

Simplify the denominator.

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Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

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

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

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

Multiply $- - \sqrt{3}$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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Reorder terms.

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

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 $- 1$ by $1$ .

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

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

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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Rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

Add $1$ and $3$ .

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

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

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

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

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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 $2$ .

$- 2 - - \sqrt{3}$

Multiply $- - \sqrt{3}$ .

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Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

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

$- 2 + \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$

Expand Using Sum/Difference Formulas tan(165)

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