To solve $\tan\left(-\frac{\pi}{2}\right)$, let's break down the process:

1. Identify the angle: $-\frac{\pi}{2} \text{ radians}$

2. Understand the angle position: The angle $-\frac{\pi}{2}$ corresponds to a clockwise rotation of $\frac{\pi}{2}$ radians (or 90 degrees) from the positive x-axis.

3. Evaluate the tangent at this angle: The tangent function is undefined at $\theta = \pm \frac{\pi}{2}$ because the tangent of an angle is the ratio of the sine to the cosine, and $\cos\left(\pm\frac{\pi}{2}\right) = 0$. This makes the tangent function undefined since division by zero is not allowed.

So, the solution is: $\tan\left(-\frac{\pi}{2}\right) \text{ is undefined.}$

Would you like more details on any step, or do you have any questions?

Here are 5 related questions you might find interesting:

1. Why is the tangent function undefined at $\theta = \frac{\pi}{2}$?
2. What is $\sin\left(-\frac{\pi}{2}\right)$?
3. How do you graph the tangent function over one period?
4. What are the values of $\tan(\theta)$ at key angles like $0$, $\frac{\pi}{4}$, and $\frac{\pi}{2}$?
5. How does the tangent function behave as $\theta$ approaches $\frac{\pi}{2}$?

Tip: When solving trigonometric functions, remember that $\tan(\theta)$ is undefined where $\cos(\theta) = 0$.