Which value is a solution for the equation [tex]\(\cot \frac{x}{2}=1\)[/tex]?

A. [tex]\(\frac{\pi}{4}\)[/tex]
B. [tex]\(\frac{7 \pi}{4}\)[/tex]
C. [tex]\(\frac{5 \pi}{2}\)[/tex]
D. [tex]\(\frac{3 \pi}{4}\)[/tex]



Answer :

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]