Why Mathematica does not simplify Sinh[ArcCosh[x]]

The post explores why Mathematica does not simplify Sinh[ArcCosh[x]] to √(x²−1), revealing that the unsimplified form is actually correct due to branch cut considerations. Defining ArcCosh rigorously requires a branch cut along (−∞, 1] in the complex plane, and approaching values on the cut from above vs. below yields different results, which explains Mathematica's seemingly verbose output.