Expand Using Sum/Difference Formulas cot(-(5pi)/12)

$\cot (- \frac{5 π}{12})$

Step 1

Replace $\cot (- \frac{5 π}{12})$ with an equivalent expression $\frac{1}{\tan (- \frac{5 π}{12})}$ using the fundamental identities.

$\frac{1}{\tan (- \frac{5 π}{12})}$

Step 2

Use a sum or difference formula on the denominator.

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $- \frac{5 π}{12}$ can be split into $\frac{π}{3} - \frac{3 π}{4}$ .

$\frac{1}{\tan (\frac{π}{3} - \frac{3 π}{4})}$

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

$\frac{1}{\frac{\tan (\frac{π}{3}) - \tan (\frac{3 π}{4})}{1 + \tan (\frac{π}{3}) \tan (\frac{3 π}{4})}}$

Remove parentheses.

$\frac{1}{\frac{\tan (\frac{π}{3}) - \tan (\frac{3 π}{4})}{1 + \tan (\frac{π}{3}) \tan (\frac{3 π}{4})}}$

Simplify the numerator.

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

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

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.

$\frac{1}{\frac{\sqrt{3} + \tan (\frac{π}{4})}{1 + \tan (\frac{π}{3}) \tan (\frac{3 π}{4})}}$

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Multiply $- (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$\frac{1}{\frac{\sqrt{3} + 1}{1 + \tan (\frac{π}{3}) \tan (\frac{3 π}{4})}}$

Multiply $- 1$ by $- 1$ .

$\frac{1}{\frac{\sqrt{3} + 1}{1 + \tan (\frac{π}{3}) \tan (\frac{3 π}{4})}}$

Simplify the denominator.

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

$\frac{1}{\frac{\sqrt{3} + 1}{1 + \sqrt{3} \tan (\frac{3 π}{4})}}$

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.

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

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Multiply $- 1$ by $1$ .

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

Move $- 1$ to the left of $\sqrt{3}$ .

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

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

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

Reorder terms.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

Factor $2$ out of $4$ .

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

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

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

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Step 3

Simplify.

Multiply $\frac{1}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

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

Multiply $\frac{1}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

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

Expand the denominator using the FOIL method.

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

Simplify.

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

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

$- 2 + \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$

Expand Using Sum/Difference Formulas cot(-(5pi)/12)

Related Questions

Support LCMGCF.com by disabling your adblocker.