To find the solution for the equation [tex]\(\cot \frac{x}{2} = 0\)[/tex], let's carefully analyze the behavior of the cotangent function.
1. Understanding [tex]\(\cot \theta\)[/tex]:
- The cotangent function, [tex]\(\cot \theta\)[/tex], is defined as [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
- [tex]\(\cot \theta\)[/tex] is zero when [tex]\(\tan \theta\)[/tex] approaches infinity, meaning that [tex]\(\tan \theta\)[/tex] should be undefined or have vertical asymptotes.
2. Setting up the Equation:
- The equation becomes [tex]\(\cot \frac{x}{2} = 0\)[/tex]. This implies that [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex] must be undefined.
- [tex]\(\tan \theta\)[/tex] is undefined when [tex]\(\theta = \frac{\pi}{2} + n\pi\)[/tex], where [tex]\(n\)[/tex] is any integer.
3. Determining Specific Solutions:
- For [tex]\(\cot \frac{x}{2} = 0\)[/tex], [tex]\(\frac{x}{2} = \frac{\pi}{2} + n\pi\)[/tex].
- Solving for [tex]\(x\)[/tex], we get [tex]\(x = \pi + 2n\pi = (2n+1)\pi\)[/tex].
4. Matching Given Options to [tex]\(x\)[/tex]:
- We need to check which of the given options satisfy this:
[tex]\[
\begin{aligned}
\text{A. } & \frac{3\pi}{4} \implies \frac{x}{2} = \frac{3\pi}{8} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\
\text{B. } & \frac{3\pi}{2} \implies \frac{x}{2} = \frac{3\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\
\text{C. } & \frac{\pi}{2} \implies \frac{x}{2} = \frac{\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\
\text{D. } & 3\pi \implies \frac{x}{2} = \frac{3\pi}{2} & \text{(Matches \(\frac{3\pi}{2} = \frac{\pi}{2} + \pi\))}
\end{aligned}
\][/tex]
Choice D ([tex]\(3\pi\)[/tex]) satisfies the given equation. Therefore, the solution to the equation [tex]\(\cot \frac{x}{2} = 0\)[/tex] is:
[tex]\[
\boxed{3\pi}
\][/tex]
