To solve the expression [tex]\( \cot \left(\frac{\pi}{2}\right) \)[/tex], we will consider the definition of the cotangent function and the value of the angle provided.
1. Recall the definition of cotangent:
The cotangent of an angle [tex]\( x \)[/tex] is the reciprocal of the tangent of [tex]\( x \)[/tex]:
[tex]\[
\cot(x) = \frac{1}{\tan(x)}
\][/tex]
2. Evaluate the tangent at the given angle:
The angle provided is [tex]\( \frac{\pi}{2} \)[/tex]. The tangent of [tex]\( \frac{\pi}{2} \)[/tex] is:
[tex]\[
\tan\left(\frac{\pi}{2}\right)
\][/tex]
3. Understanding the tangent at [tex]\(\frac{\pi}{2}\)[/tex]:
The tangent function [tex]\( \tan(x) \)[/tex] is defined as:
[tex]\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\][/tex]
At [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0
\][/tex]
Therefore:
[tex]\[
\tan\left(\frac{\pi}{2}\right) = \frac{1}{0}
\][/tex]
The expression [tex]\(\frac{1}{0}\)[/tex] is undefined, leading many to consider [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex] as undefined because division by zero is undefined.
4. Determine the cotangent:
Given that [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex] is undefined:
[tex]\[
\cot\left(\frac{\pi}{2}\right) = \frac{1}{\tan\left(\frac{\pi}{2}\right)}
\][/tex]
As per the definition, [tex]\( \tan\left(\frac{\pi}{2}\right) \)[/tex] is infinity, and thus [tex]\(\cot\left(\frac{\pi}{2}\right)\)[/tex] would be:
[tex]\[
\cot\left(\frac{\pi}{2}\right) = 0
\][/tex]
Hence, based on the analysis, the answer is:
[tex]\[
\boxed{0}
\][/tex]
So, the correct choice from the options is B. 0
