Find the Exact Value cot((13pi)/8)

$\cot (\frac{13 π}{8})$

Step 1

Rewrite $\frac{13 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{\frac{13 π}{4}}{2})$

Step 2

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{\frac{13 π}{4}}{2})}$

Step 3

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

Step 4

Change the $\pm$ to $-$ because cotangent is negative in the fourth quadrant.

$\frac{1}{- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

Step 5

Simplify $\frac{1}{- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$ .

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

Move the negative in front of the fraction.

$- \frac{1}{\sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

Simplify the numerator.

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \frac{1}{\sqrt{\frac{1 - \cos (\frac{5 π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

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

$- \frac{1}{\sqrt{\frac{1 - - \cos (\frac{π}{4})}{1 + \cos (\frac{13 π}{4})}}}$

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{1}{\sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}}}$

Multiply $- - \frac{\sqrt{2}}{2}$ .

Multiply $- 1$ by $- 1$ .

$- \frac{1}{\sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}}}$

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$- \frac{1}{\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}}}$

Write $1$ as a fraction with a common denominator.

$- \frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}}}$

Combine the numerators over the common denominator.

$- \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}}}$

Simplify the denominator.

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (\frac{5 π}{4})}}}$

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

$- \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 - \cos (\frac{π}{4})}}}$

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the denominator.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

$- \frac{1}{\sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}}}$

Simplify.

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\sqrt{2}$ and $\frac{2 + \sqrt{2}}{2}$ .

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

Find the common denominator.

Write $2$ as a fraction with denominator $1$ .

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

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Write $\sqrt{2}$ as a fraction with denominator $1$ .

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

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $2$ by $2$ .

$- \frac{1}{\sqrt{\frac{4 + \sqrt{2} \cdot 2 + \sqrt{2} (2 + \sqrt{2})}{2}}}$

Move $2$ to the left of $\sqrt{2}$ .

$- \frac{1}{\sqrt{\frac{4 + 2 \cdot \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}}}$

Apply the distributive property.

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

Move $2$ to the left of $\sqrt{2}$ .

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

Combine using the product rule for radicals.

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

Simplify each term.

Multiply $2$ by $2$ .

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

Rewrite $4$ as $2^{2}$ .

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

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

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

Add $4$ and $2$ .

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

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

$- \frac{1}{\sqrt{\frac{6 + 4 \sqrt{2}}{2}}}$

Cancel the common factor of $6 + 4 \sqrt{2}$ and $2$ .

Factor $2$ out of $6$ .

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

Factor $2$ out of $4 \sqrt{2}$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.41421356 \dots$

Find the Exact Value cot((13pi)/8)

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