The expression $\cot\left(\frac{\pi}{2} + x\right)$ can be simplified using trigonometric identities. Specifically, we can use the cotangent addition identity for angles of the form $\frac{\pi}{2} + x$.

Cotangent Identity for $\frac{\pi}{2}$-Shift:

$\cot\left(\frac{\pi}{2} + x\right) = -\tan(x)$

So, the simplified form of $\cot\left(\frac{\pi}{2} + x\right)$ is: $\cot\left(\frac{\pi}{2} + x\right) = -\tan(x)$

Final Answer:

$\boxed{-\tan(x)}$

Would you like further explanation or details?

Here are 5 related questions to consider:

1. What is the general rule for the cotangent of a sum of angles?
2. How does $\cot(\pi - x)$ compare with $\cot(\pi + x)$?
3. Can you derive the identity $\cot\left(\frac{\pi}{2} + x\right)$ using sine and cosine definitions?
4. How does the cotangent identity change for other phase shifts, such as $\cot(\pi/2 - x)$?
5. What is the graphical interpretation of $\cot(x)$ and $\tan(x)$?

Tip: For angle transformations, knowing the fundamental trigonometric identities for $\frac{\pi}{2}$-shifts is key to simplifying many trigonometric expressions.