[math-fun] wrongish looking elementary identity

ArcTan[(Tan[π/√2] - Tanh[π/√2])/( Tan[π/√2] + Tanh[π/√2])] == Arg[Sinh[π √I]] - π/4 FullSimplify fails. FullSimplify[Tan/@%] fails. But then FullSimplify@TrigExpand@% says True. Imagine if FullSimplify[ ArcTan[(Tan[π/√2] - Tanh[π/√2])/(Tan[π/√2] + Tanh[π/√2])] + π/4] gave Arg[Sinh[π √I]] because it has a (much) lower ByteCount! Who could possibly object?-) --rwg