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To solve this problem, let's go through the steps.

### Step 1: Understand the Given Information
We know:
1. \(\cot x = -\frac{1}{2}\)
2. \(x\) is in the 4th quadrant.

In the 4th quadrant:
- \(\cos x\) is positive.
- \(\sin x\) is negative.

### Step 2: Relate \(\cot x\) to \(\sin x\) and \(\cos x\)
Since \(\cot x = \frac{\cos x}{\sin x}\), we can set up the equation:
\[
\cot x = -\frac{1}{2} = \frac{\cos x}{\sin x}.
\]

Let’s assume \(\cos x = k\) and \(\sin x = -2k\) (keeping the signs consistent with quadrant IV).

### Step 3: Use the Pythagorean Identity
Since \(\sin^2 x + \cos^2 x = 1\), substitute \(\sin x = -2k\) and \(\cos x = k\):
\[
(-2k)^2 + k^2 = 1.
\]
\[
4k^2 + k^2 = 1.
\]
\[
5k^2 = 1.
\]
\[
k^2 = \frac{1}{5}.
\]
\[
k = \frac{1}{\sqrt{5}} \text{ or } k = -\frac{1}{\sqrt{5}}.
\]

Since \(\cos x\) is positive in the 4th quadrant, we take \(k = \frac{1}{\sqrt{5}}\). Thus:
\[
\cos x = \frac{1}{\sqrt{5}} \quad \text{and} \quad \sin x = -\frac{2}{\sqrt{5}}.
\]

### Step 4: Use the Double-Angle Formula for \(\sin 2x\)
The formula for \(\sin 2x\) is:
\[
\sin 2x = 2 \sin x \cos x.
\]

Substitute \(\sin x = -\frac{2}{\sqrt{5}}\) and \(\cos x = \frac{1}{\sqrt{5}}\):
\[
\sin 2x = 2 \left(-\frac{2}{\sqrt{5}}\right) \left(\frac{1}{\sqrt{5}}\right).
\]
\[
\sin 2x = 2 \cdot -\frac{2}{5} = -\frac{4}{5}.
\]

### Conclusion
The value of \(\sin 2x\) is \(-\frac{4}{5}\).

### Answer
The correct answer is:
**(a) \(-\frac{4}{5}\)**.

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Would you like a deeper explanation of any step, or do you have further questions?

### Related Questions
1. What are the values of \(\cos 2x\) and \(\tan 2x\) given the same conditions?
2. How would the solution change if \(x\) were in the 2nd quadrant?
3. How is the double-angle formula for cosine derived?
4. Why does the sign of trigonometric functions vary across quadrants?
5. How does \(\cot x\) relate to \(\tan x\) and what transformations are involved?

### Tip
When solving trigonometric problems, always pay attention to the quadrant since it affects the signs of sine, cosine, and tangent.

If cot x = -1/2 and x is 4th quadrant, then find sin 2x.

Learn how to find sin 2x given cot x = -1/2 with x in the 4th quadrant. This trigonometry problem covers key concepts like the Pythagorean identity and the double angle formula. Ideal for students in Grades 10-12.

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