Mike Hirschhorn (using Maple) solved both puzzles off list. Rich>asin(tanh(x))-atan(sinh(x)) is even flatter! --- Quoting Bill Gosper <billgosper@gmail.com>:

From Neil Bickford (under protest): Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes berserk at π/2. Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 Berserk: Local minima and maxima cluster at π/2, vibrating between roughly .5 and 2.5. What limits, if any, do they approach? Note that by analytic continuation, an arbitrarily short segment of the flat part near zero completely predicts the berserk part west of π/2. Are all the terms of the series approximated above positive?

Plot[ArcSinh[Tan[y]] - ArcTanh[Sin[y]], {y, -2, 2}, PlotRange -> All] looks that flat for awhile. Then, a Big Bang violates analytic continuation. So THAT's what caused it! --rwg