1. The problem is to simplify the expression $\frac{e^x}{2} + \frac{e^{-x}}{2}$.\n\n2. We start by recognizing that the expression is a sum of two terms, each divided by 2. We can write it as:\n$$\frac{e^x + e^{-x}}{2}$$\n\n3. This expression matches the definition of the hyperbolic cosine function $\cosh x$, which is defined as:\n$$\cosh x = \frac{e^x + e^{-x}}{2}$$\n\n4. Therefore, the simplified form of the expression is:\n$$\cosh x$$\n\n5. In plain language, the sum of $e^x$ and $e^{-x}$ divided by 2 is exactly the hyperbolic cosine of $x$. This function is useful in many areas of mathematics and physics, especially in solving differential equations and describing certain curves.