The expression you're evaluating is:

$\tan\left(\frac{\pi}{2}\right)$

The tangent function is defined as the ratio of sine to cosine:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

At the angle $\frac{\pi}{2}$ (or 90°), we have:

$\sin\left(\frac{\pi}{2}\right) = 1$ $\cos\left(\frac{\pi}{2}\right) = 0$

Since the denominator of the tangent function is $\cos\left(\frac{\pi}{2}\right) = 0$, the expression for $\tan\left(\frac{\pi}{2}\right)$ is undefined.

Therefore, the value of $\tan\left(\frac{\pi}{2}\right)$ is undefined.

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Would you like further clarification or assistance with another topic? Here are some related questions to explore:

1. What happens to the tangent function at other quadrantal angles like $0$ or $\pi$?
2. How is the secant function related to the cosine function at quadrantal angles?
3. What is the behavior of trigonometric functions near vertical asymptotes, like at $\frac{\pi}{2}$?
4. How do we find the values of sine and cosine at key angles like $\frac{\pi}{4}$?
5. Can the tangent function ever be defined when the cosine function is 0?

Tip: Always remember that trigonometric functions involving division can become undefined if the denominator is zero!