If `cos theta= -sqrt(3)/2` and `pi < theta < (3pi)/2` then find the value of `4tan^2 theta- 3cosec^2 theta`

Explanation

To solve the problem, we need to find the value of \(4\tan^2 \theta - 3\csc^2 \theta\) given that \(\cos \theta = -\frac{\sqrt{3}}{2}\) and \(\pi < \theta < \frac{3\pi}{2}\).

### Step 1: Identify the quadrant and values of trigonometric functions
Since \(\theta\) is in the third quadrant (where \(\pi < \theta < \frac{3\pi}{2}\)), we know that:
- \(\cos \theta\) is negative.
- \(\sin \theta\) is also negative.

Given \(\cos \theta = -\frac{\sqrt{3}}{2}\), we can use the Pythagorean identity to find \(\sin \theta\).
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