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.

Learn how to simplify Cot(π/2 + x) using trigonometric identities. This solution involves key concepts such as the cotangent identity and angle addition. Suitable for students in Grades 9-12.

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