Evaluate (cot((3pi)/2)-cot(pi/3))/((3pi)/2-pi/3)

$\frac{\cot (\frac{3 π}{2}) - \cot (\frac{π}{3})}{\frac{3 π}{2} - \frac{π}{3}}$

Multiply the numerator and denominator of the complex fraction by $6$ .

Multiply $\frac{\cot (\frac{3 π}{2}) - \cot (\frac{π}{3})}{\frac{3 π}{2} - \frac{π}{3}}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\cot (\frac{3 π}{2}) - \cot (\frac{π}{3})}{\frac{3 π}{2} - \frac{π}{3}}$

Combine.

$\frac{6 (\cot (\frac{3 π}{2}) - \cot (\frac{π}{3}))}{6 (\frac{3 π}{2} - \frac{π}{3})}$

Apply the distributive property.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{6 \frac{3 π}{2} + 6 (- \frac{π}{3})}$

Simplify by cancelling.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{2 (3) \frac{3 π}{2} + 6 (- \frac{π}{3})}$

Cancel the common factor.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{2 \cdot 3 \frac{3 π}{2} + 6 (- \frac{π}{3})}$

Rewrite the expression.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{3 (3 π) + 6 (- \frac{π}{3})}$

Multiply $3$ by $3$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π + 6 (- \frac{π}{3})}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{π}{3}$ into the numerator.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π + 6 \frac{- π}{3}}$

Factor $3$ out of $6$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π + 3 (2) \frac{- π}{3}}$

Cancel the common factor.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π + 3 \cdot 2 \frac{- π}{3}}$

Rewrite the expression.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π + 2 (- π)}$

Multiply $- 1$ by $2$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))}{9 π - 2 π}$

Simplify the numerator.

Factor $6$ out of $6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{π}{3}))$ .

$\frac{6 (\cot (\frac{3 π}{2}) - \cot (\frac{π}{3}))}{9 π - 2 π}$

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

$\frac{6 (- \cot (\frac{π}{2}) - \cot (\frac{π}{3}))}{9 π - 2 π}$

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$\frac{6 (- 0 - \cot (\frac{π}{3}))}{9 π - 2 π}$

Multiply $- 1$ by $0$ .

$\frac{6 (0 - \cot (\frac{π}{3}))}{9 π - 2 π}$

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$\frac{6 (0 - \frac{1}{\sqrt{3}})}{9 π - 2 π}$

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

$\frac{6 (0 - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))}{9 π - 2 π}$

Combine and simplify the denominator.

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

$\frac{6 (0 - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}})}{9 π - 2 π}$

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

$\frac{6 (0 - \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}{9 π - 2 π}$

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

$\frac{6 (0 - \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}{9 π - 2 π}$

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

$\frac{6 (0 - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}})}{9 π - 2 π}$

Add $1$ and $1$ .

$\frac{6 (0 - \frac{\sqrt{3}}{{\sqrt{3}}^{2}})}{9 π - 2 π}$

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

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

$\frac{6 (0 - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}{9 π - 2 π}$

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

$\frac{6 (0 - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}{9 π - 2 π}$

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

$\frac{6 (0 - \frac{\sqrt{3}}{3^{\frac{2}{2}}})}{9 π - 2 π}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 (0 - \frac{\sqrt{3}}{3^{\frac{2}{2}}})}{9 π - 2 π}$

Rewrite the expression.

$\frac{6 (0 - \frac{\sqrt{3}}{3^{1}})}{9 π - 2 π}$

Evaluate the exponent.

$\frac{6 (0 - \frac{\sqrt{3}}{3})}{9 π - 2 π}$

Subtract $\frac{\sqrt{3}}{3}$ from $0$ .

$\frac{6 \cdot - 1 \frac{\sqrt{3}}{3}}{9 π - 2 π}$

Combine exponents.

Factor out negative.

$\frac{- (6 \frac{\sqrt{3}}{3})}{9 π - 2 π}$

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

$\frac{- \frac{6 \sqrt{3}}{3}}{9 π - 2 π}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 \sqrt{3}$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{- \frac{3 (2 \sqrt{3})}{3 (1)}}{9 π - 2 π}$

Cancel the common factor.

$\frac{- \frac{3 (2 \sqrt{3})}{3 \cdot 1}}{9 π - 2 π}$

Rewrite the expression.

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

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

$\frac{- (2 \sqrt{3})}{9 π - 2 π}$

$\frac{- 1 \cdot 2 \sqrt{3}}{9 π - 2 π}$

Multiply $- 1$ by $2$ .

$\frac{- 2 \sqrt{3}}{9 π - 2 π}$

Simplify by adding terms.

Subtract $2 π$ from $9 π$ .

$\frac{- 2 \sqrt{3}}{7 π}$

Move the negative in front of the fraction.

$- \frac{2 \sqrt{3}}{7 π}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{3}}{7 π}$

Decimal Form:

$- 0.15752254 \dots$

Evaluate (cot((3pi)/2)-cot(pi/3))/((3pi)/2-pi/3)

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