Always remember the range of inverse functions. For $\cos^{-1}x$, $\sec^{-1}x$, and $\cot^{-1}x$, the range is $[0, \pi]$ or $(0, \pi)$. If the input is negative, the answer will always be an obtuse angle (Quadrant II).
1. Setting the Equation: Let $y = \cot^{-1}(-1)$. This implies:
$$\cot y = -1$$
2. Finding the Angle: We know that $\cot(\frac{\pi}{4}) = 1$. Since the value is negative, the angle must lie in the second quadrant (as per the range $(0, \pi)$).
$$\cot y = -\cot\left(\frac{\pi}{4}\right)$$
Using the identity $\cot(\pi - \theta) = -\cot \theta$:
$$\cot y = \cot\left(\pi - \frac{\pi}{4}\right)$$
$$\cot y = \cot\left(\frac{3\pi}{4}\right)$$