Step 1: Understanding the Concept:
The problem asks to find the value of $\cos\theta$ given a trigonometric equation and the quadrant of the angle $\theta$. The first step is to simplify the given equation.
Step 2: Key Formula or Approach:
We use the reciprocal identity for cosecant: $\csc\theta = \frac{1}{\sin\theta}$. Substituting this into the given equation will simplify it.
Step 3: Detailed Explanation:
The given equation is $\cos\theta \csc\theta = -1$.
Substitute $\csc\theta = \frac{1}{\sin\theta}$:
\[ \cos\theta \left(\frac{1}{\sin\theta}\right) = -1 \]
\[ \frac{\cos\theta}{\sin\theta} = -1 \]
We know that $\frac{\cos\theta}{\sin\theta} = \cot\theta$. So,
\[ \cot\theta = -1 \]
We are given that $\theta$ lies in the second quadrant. In the second quadrant, sine is positive and cosine is negative.
If $\cot\theta = -1$, this means the reference angle for $\theta$ is one where the tangent is 1, which is $45^\circ$ or $\frac{\pi}{4}$ radians.
In the second quadrant, the angle $\theta$ would be $180^\circ - 45^\circ = 135^\circ$ (or $\pi - \frac{\pi}{4} = \frac{3\pi}{4}$).
Now, we need to find the value of $\cos\theta$ for $\theta = 135^\circ$:
\[ \cos(135^\circ) = \cos(180^\circ - 45^\circ) = -\cos(45^\circ) \]
Using the value $\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$, we get:
\[ \cos\theta = -\frac{\sqrt{2}}{2} \]
Step 4: Final Answer:
The value of $\cos\theta$ is $-\frac{\sqrt{2}}{2}$. Therefore, option (C) is the correct answer. Options B and D are incorrect as cosine must be negative in the second quadrant, and its value cannot be less than -1.