Given information:
- Angle theta is between 180 degrees and 270 degrees.
- The equation satisfied by theta is 9cot^2(theta) - 4cosec^2(theta) = 1.

To find:
The value of sin(2theta) - cot(2theta).

Solution:
Let's break down the problem into smaller steps:

Step 1: Determine the value of cot(theta) and cosec(theta):
We know that cot(theta) = 1/tan(theta) and cosec(theta) = 1/sin(theta). Let's find the values of these trigonometric ratios.

Since theta is between 180 degrees and 270 degrees, it lies in the third quadrant of the unit circle. In the third quadrant, sin(theta) is negative, and cos(theta) and tan(theta) are positive.

Using the unit circle, we can determine that sin(theta) = -sqrt(2)/2 and cos(theta) = -sqrt(2)/2.

Therefore, cot(theta) = 1/tan(theta) = 1/(-sqrt(2)/2) = -sqrt(2).
And cosec(theta) = 1/sin(theta) = 1/(-sqrt(2)/2) = -2/sqrt(2) = -sqrt(2).

Step 2: Substitute the values of cot(theta) and cosec(theta) into the equation:
We are given the equation 9cot^2(theta) - 4cosec^2(theta) = 1. Substituting the values of cot(theta) and cosec(theta), we get:

9(-sqrt(2))^2 - 4(-sqrt(2))^2 = 1
9(2) - 4(2) = 1
18 - 8 = 1
10 = 1

This equation is not true, which means there is no solution for theta that satisfies the given equation.

Step 3: Find sin(2theta) and cot(2theta) using double angle formulas:
Since there is no solution for the given equation, we cannot directly find the value of sin(2theta) - cot(2theta).

However, we can still find the values of sin(2theta) and cot(2theta) using double angle formulas:

sin(2theta) = 2sin(theta)cos(theta)
cot(2theta) = cot^2(theta) - 1

Step 4: Substitute the values of sin(theta) and cos(theta) into the double angle formulas:
Using the values of sin(theta) = -sqrt(2)/2 and cos(theta) = -sqrt(2)/2, we can find sin(2theta) and cot(2theta).

sin(2theta) = 2sin(theta)cos(theta) = 2(-sqrt(2)/2)(-sqrt(2)/2) = 2(2/2) = 2/2 = 1
cot(2theta) = cot^2(theta) - 1 = (-sqrt(2))^2 - 1 = 2 - 1 = 1

Step 5: Calculate sin(2theta) - cot(2theta):
Now that we have the values of