Find the Exact Value cot(-315)

$\cot (- 315)$

Step 1

Rewrite $- 315$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{- 630}{2})$

Step 2

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{- 630}{2})}$

Step 3

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 630)}{1 + \cos (- 630)}}}$

Step 4

Change the $\pm$ to $+$ because cotangent is positive in the first quadrant.

$\frac{1}{\sqrt{\frac{1 - \cos (- 630)}{1 + \cos (- 630)}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 630)}{1 + \cos (- 630)}}}$ .

Simplify the numerator.

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{1}{\sqrt{\frac{1 - \cos (90)}{1 + \cos (- 630)}}}$

The exact value of $\cos (90)$ is $0$ .

$\frac{1}{\sqrt{\frac{1 - 0}{1 + \cos (- 630)}}}$

Multiply $- 1$ by $0$ .

$\frac{1}{\sqrt{\frac{1 + 0}{1 + \cos (- 630)}}}$

Add $1$ and $0$ .

$\frac{1}{\sqrt{\frac{1}{1 + \cos (- 630)}}}$

Simplify the denominator.

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{1}{\sqrt{\frac{1}{1 + \cos (90)}}}$

The exact value of $\cos (90)$ is $0$ .

$\frac{1}{\sqrt{\frac{1}{1 + 0}}}$

Add $1$ and $0$ .

$\frac{1}{\sqrt{\frac{1}{1}}}$

Simplify the denominator.

Divide $1$ by $1$ .

$\frac{1}{\sqrt{1}}$

Any root of $1$ is $1$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$