When you find a simple trigonometric ratio like \(\cot \theta = -1\), you can quickly visualize the angle. Cotangent is the ratio \(x/y\). For it to be -1, \(x = -y\). In the second quadrant, x is negative and y is positive, which fits. This corresponds to the line \(y = -x\) in the second quadrant, which makes a 45\(^{\circ}\) angle with the negative x-axis. The coordinates on the unit circle would be \((-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})\), so \(\cos \theta\) is the x-coordinate, which is \(-\frac{1}{\sqrt{2}}\) or \(-\frac{\sqrt{2}}{2}\).