To determine which value of [tex]\( x \)[/tex] satisfies the equation [tex]\(\cot \frac{x}{2} = 1\)[/tex], we need to analyze each of the given options individually.
Recall that [tex]\(\cot \theta = 1\)[/tex] if and only if [tex]\(\theta = \frac{\pi}{4} + k\pi\)[/tex] for some integer [tex]\( k \)[/tex]. This is because [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex], and [tex]\(\tan \theta = 1\)[/tex] when [tex]\(\theta = \frac{\pi}{4} + k\pi\)[/tex].
Let's check each given option:
Option A: [tex]\(\frac{\pi}{4}\)[/tex]
[tex]\[
\frac{\pi}{4} \quad \text{implies that} \quad \frac{x}{2} = \frac{\pi}{4}
\][/tex]
Multiplying both sides by 2:
[tex]\[
x = \frac{\pi}{2}
\][/tex]
Now, check [tex]\(\cot \frac{x}{2} = \cot \frac{\pi}{4} = 1\)[/tex]. Hence, this seems to work.
Option B: [tex]\(\frac{7 \pi}{4}\)[/tex]
[tex]\[
\frac{7 \pi}{4} \quad \text{implies that} \quad \frac{x}{2} = \frac{7 \pi}{4}
\][/tex]
Multiplying both sides by 2:
[tex]\[
x = \frac{7 \pi}{2}
\][/tex]
Now, check [tex]\(\cot \frac{x}{2} = \cot \frac{7 \pi}{4}\)[/tex]. Since [tex]\(\frac{7 \pi}{4}\)[/tex] is not an odd multiple of [tex]\(\frac{\pi}{4}\)[/tex], it does not work for [tex]\(\cot \frac{x}{2} = 1\)[/tex].
Option C: [tex]\(\frac{5 \pi}{2}\)[/tex]
[tex]\[
\frac{5 \pi}{2} \quad \text{implies that} \quad \frac{x}{2} = \frac{5 \pi}{2}
\][/tex]
Multiplying both sides by 2:
[tex]\[
x = 5\pi
\][/tex]
Now check [tex]\(\cot \frac{x}{2} = \cot \frac{5 \pi}{2}\)[/tex]:
[tex]\(\frac{5\pi}{2}\)[/tex] simplifies to [tex]\(\pi + \frac{\pi}{2}\)[/tex], which is an odd multiple of [tex]\(\frac{\pi}{4}\)[/tex]. Hence:
[tex]\[
\cot \frac{5\pi}{4} = \cot \left(\pi + \frac{\pi}{4}\right) = \cot \frac{\pi}{4} = 1
\][/tex]
Thus, this is indeed correct.
Option D: [tex]\(\frac{3 \pi}{4}\)[/tex]
[tex]\[
\frac{3 \pi}{4} \quad \text{implies that} \quad \frac{x}{2} = \frac{3 \pi}{4}
\][/tex]
Multiplying both sides by 2:
[tex]\[
x = \frac{3 \pi}{2}
\][/tex]
Now check [tex]\(\cot \frac{x}{2} = \cot \frac{3 \pi}{4}\)[/tex]. Since [tex]\(\frac{3 \pi}{4}\)[/tex] is not an odd multiple of [tex]\(\frac{\pi}{4}\)[/tex], it does not work for [tex]\(\cot \frac{x}{2} = 1\)[/tex].
Conclusion:
Among the options, [tex]\( x = \frac{5 \pi}{2} \)[/tex] satisfies the equation [tex]\(\cot \frac{x}{2} = 1\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{5\pi}{2}}
\][/tex]
