To determine which value of [tex]\( x \)[/tex] satisfies the equation [tex]\(\cot \left( \frac{x}{2} \right) = -1\)[/tex], we need to analyze each of the given choices and check whether they satisfy the provided condition.
We will go through each option step-by-step.
1. Option A: [tex]\( x = \frac{3\pi}{4} \)[/tex]
[tex]\[
\text{Calculate} \ \frac{x}{2} = \frac{\frac{3\pi}{4}}{2} = \frac{3\pi}{8}
\][/tex]
[tex]\(\cot \left( \frac{3\pi}{8} \right)\)[/tex] does not equal to -1.
2. Option B: [tex]\( x = \frac{7\pi}{4} \)[/tex]
[tex]\[
\text{Calculate} \ \frac{x}{2} = \frac{\frac{7\pi}{4}}{2} = \frac{7\pi}{8}
\][/tex]
[tex]\(\cot \left( \frac{7\pi}{8} \right)\)[/tex] does not equal to -1.
3. Option C: [tex]\( x = \frac{3\pi}{2} \)[/tex]
[tex]\[
\text{Calculate} \ \frac{x}{2} = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4}
\][/tex]
[tex]\(\cot \left( \frac{3\pi}{4} \right) = \cot \left( 135^\circ \right) = -1\)[/tex].
Thus, [tex]\(\cot \left( \frac{3\pi}{4} \right) = -1\)[/tex].
4. Option D: [tex]\( x = \frac{5\pi}{4} \)[/tex]
[tex]\[
\text{Calculate} \ \frac{x}{2} = \frac{\frac{5\pi}{4}}{2} = \frac{5\pi}{8}
\][/tex]
[tex]\(\cot \left( \frac{5\pi}{8} \right)\)[/tex] does not equal to -1.
From these calculations, we can see that the only [tex]\( x \)[/tex] value that satisfies [tex]\(\cot \left( \frac{x}{2} \right) = -1\)[/tex] is:
[tex]\[
\boxed{\frac{3\pi}{2}}
\][/tex]
