If `costheta=1/2` and `pi< theta < (3pi)/2` , find the value of `4tan^2theta-3cos e c^2thetadot`

Explanation

To solve the problem, we need to find the value of the expression \( 4\tan^2\theta - 3\csc^2\theta \) given that \( \cos\theta = \frac{1}{2} \) and \( \pi < \theta < \frac{3\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the value of \(\cos\theta\)**:
Given that \( \cos\theta = \frac{1}{2} \), we can determine the corresponding angle. In the interval \( \pi < \theta < \frac{3\pi}{2} \), \(\theta\) must be \( \frac{5\pi}{3} \) or \( \frac{7\pi}{3} \), but since we are in the third quadrant, we take \( \theta = \frac{5\pi}{3} \).

2. **Calculate \(\sec\theta\)**:
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