Answer
$-1$
Work Step by Step
RECALL:
(1)The reference angle of a given angle is equal to the smallest acute angle that the terminal side makes with the x-axis.
(2) Based on the location of the terminal side of an angle $\theta$, the reference angle can be found using the formula:
(i) Quadrant I: $\theta$
(ii) Quadrant II: $\pi-\theta$
(iii) Quadrant III: $\theta-\pi$
(iv) Quadrant IV: $2\pi-\theta$
The given angle is in Quadrant IV so its reference angle is:
$=2\pi - \dfrac{7\pi}{4}
\\=\dfrac{\pi}{4}$
The value of cotangent in Quadrant IV is negative.
Thus,
$\cot{\frac{7\pi}{4}} = -\cot{\frac{\pi}{4}}$
RECALL:
$\cot{\theta} = \dfrac{1}{\tan{\theta}}$
Thus,
$-\cot{\frac{\pi}{4}} = -\dfrac{1}{\tan{\frac{\pi}{4}}}$
$\dfrac{\pi}{4}$ is a special angle whose tangent value is $1$.
Thus,
$\cot{\dfrac{7\pi}{4}}=-\dfrac{1}{1}=-1$
