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

Mathematica returns an unexpected form for Sinh[ArcCosh[x]] instead of the simpler √(x²−1). The reason is that the simpler form is only valid when x ≥ −1, while Mathematica's result is correct for all complex inputs. The post explains how ArcCosh requires a branch cut in the complex plane (from −∞ to +1), and how the square root function similarly requires careful definition via branch cuts and analytic continuation. For x = 2, Sinh[ArcCosh[2]] correctly returns −√3, not √3. If you only care about real x ≥ −1, you can pass an assumption to Simplify and get the expected simpler form.