To determine the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(\cot \theta = -\frac{\sqrt{5}}{2}\)[/tex] and [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], we can follow these steps:
1. Understand the given information:
[tex]\(\cot \theta = -\frac{\sqrt{5}}{2}\)[/tex]
The cotangent function [tex]\(\cot \theta\)[/tex] is the reciprocal of the tangent function [tex]\(\tan \theta\)[/tex]. Thus, we have:
[tex]\[
\cot \theta = \frac{1}{\tan \theta}
\][/tex]
2. Express [tex]\(\tan \theta\)[/tex] in terms of [tex]\(\cot \theta\)[/tex]:
Given [tex]\(\cot \theta\)[/tex], we can find [tex]\(\tan \theta\)[/tex] by taking the reciprocal of [tex]\(\cot \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{1}{\cot \theta}
\][/tex]
3. Substitute the given value of [tex]\(\cot \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{1}{-\frac{\sqrt{5}}{2}} = \frac{2}{-\sqrt{5}}
\][/tex]
4. Simplify the expression:
Simplify [tex]\(\tan \theta\)[/tex] by rationalizing the denominator:
[tex]\[
\tan \theta = \frac{2}{-\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \cdot (-1) = -\frac{2\sqrt{5}}{5}
\][/tex]
Therefore, the correct value of [tex]\(\tan \theta\)[/tex] is [tex]\(\boxed{-\frac{2\sqrt{5}}{5}}\)[/tex], which matches option c.
