(*SetOptions[EvaluationNotebook[],FontSize->20]*)b=TrigToExp[((2 s+1) Coth[a (s+1/2)]-Coth[a/2]) Sinh[a/2]/Sinh[a (s+1/2)]]//Expand//Simplify (*//Factor*)
n=b//Numerator
d=b//Denominator

nn = n/E^(a+2 a s)//ExpandAll // ExpToTrig
dd = d/E^(a+2 a s)// ExpandAll // ExpToTrig
nn/dd // FullSimplify // Factor // TraditionalForm