To simplify the expression $\frac{\cos(x)}{2i}$, we can approach it in the context of complex numbers and Euler's formula.

First, let's understand the relationship between the cosine function and the complex exponential function. Using Euler's formula:

$e^{ix} = \cos(x) + i\sin(x)$

This means:

$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$

Now, substitute this into the original expression:

$\frac{\cos(x)}{2i} = \frac{\frac{e^{ix} + e^{-ix}}{2}}{2i} = \frac{e^{ix} + e^{-ix}}{4i}$

Thus, the simplified form of the expression $\frac{\cos(x)}{2i}$ is:

$\frac{e^{ix} + e^{-ix}}{4i}$

Would you like more details or have any questions on this? Here are five related questions to consider:

1. How does Euler's formula help in simplifying trigonometric expressions?
2. What is the connection between complex numbers and trigonometry?
3. How can you express sine functions in terms of exponentials?
4. What does the term $i$ represent in complex numbers?
5. Can the expression $\frac{\cos(x)}{2i}$ be simplified further in certain contexts?

Tip: When dealing with complex exponentials, remember that they often provide simpler ways to represent trigonometric functions.