Clear[f]

f[b_, l_] =  FullSimplify[D[ Sinh[b(l+1/2)]/Sinh[b/2], b]] ;
f[b, l] // TraditionalForm

f[b, 1] // FullSimplify
f[b, 2] // FullSimplify // Factor
f[b, 3] // FullSimplify // Factor

TrigExpand[ Sinh[x + 1] - Sinh[x]] // Factor

Clear[x]
Plot[ 
{
x Sinh[ x + 1] - (x+1) Sinh[x],
Sinh[x + 1/2],
x E^x/2(*,  
Sqrt[E]E^x/2*)
}, 
{x, 0, 4},
AxesLabel -> {x},
PlotLegends->Placed[{
x sinh[x + 1] - (x + 1)sinh[x], 
sinh[x+(1/2)], 
x E^x/2(*, 
 Sqrt[E] E^x/(2)*)
}, Center]
]

Manipulate[
Plot[ 
{
Sinh[1/x]/(E^( a/x) + 1 + 2 Cosh[1/x])

(*,
Sinh[1/x]/(E^( a/x) + 1),
Sinh[1/x]/( 1 + 2 Cosh[1/x])*)
}
, 
{x, 0.001, 100},
PlotRange -> Full,
AxesLabel -> {(Subscript[k, B])T/(μ ℏ ℰ), "<(Subscript[L, z])>/2 ℏ"}
]
, {{a, 16}, 0.001, 100}]

Manipulate[
Plot[ 
{
Sinh[1/x]/(E^( a/x) + 1 + 2 Cosh[1/x])
, x/(4 x^2 + a x + 1)
, 1/(4 x)

}
, 
{x, 0.001, 100},
AxesLabel -> {(Subscript[k, B])T/(μ ℏ ℰ)}
]
, {{a, 16}, 0.001, 100}]

Manipulate[
Plot[
{
1/ (1 + E^(a/t))
(*,  1/ (1 + E^(-a/t))*)
},{t,0,10}
],
{a, 1, 10}
]