To determine the value of [tex]\(\tan \left(\frac{\pi}{2}\right)\)[/tex], let's first recall the definition of the tangent function and properties of special angles.
The tangent function, [tex]\(\tan(\theta)\)[/tex], is defined as the ratio of the sine and cosine functions:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Now, let's plug in [tex]\(\theta = \frac{\pi}{2}\)[/tex]:
[tex]\[
\tan \left(\frac{\pi}{2}\right) = \frac{\sin \left( \frac{\pi}{2} \right)}{\cos \left( \frac{\pi}{2} \right)}
\][/tex]
We need to know the sine and cosine of [tex]\(\frac{\pi}{2}\)[/tex]:
[tex]\[
\sin \left( \frac{\pi}{2} \right) = 1
\][/tex]
[tex]\[
\cos \left( \frac{\pi}{2} \right) = 0
\][/tex]
Substituting these values into the equation:
[tex]\[
\tan \left( \frac{\pi}{2} \right) = \frac{1}{0}
\][/tex]
The division by zero is undefined in mathematics. Therefore, the value of [tex]\(\tan \left(\frac{\pi}{2}\right)\)[/tex] is:
[tex]\[
\boxed{\text{Undefined}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{B. Undefined}
\][/tex]
