In[]:=
Clear[ϕ];Clear[L];Clear[llog];Clear[m1];Clear[m2];Clear[m3];
In[]:=
P1[ϕ_]:=​​2^-4*ExpToTrig[((E^(I*(ϕ/2))+​​E^(-I*(ϕ/2)))(E^(I*(-(ϕ/2)))+​​E^(I*(ϕ/2))))]^2;​​FullSimplify[P1[ϕ]]
Out[]=
4
Cos
ϕ
2

In[]:=
P2[ϕ_]:=2^-3*ExpToTrig[(E^(I*(ϕ))-E^(-I*(ϕ)))(E^(I*(-ϕ))-E^(I*(ϕ)))];​​FullSimplify[P2[ϕ]]
Out[]=
2
Sin[ϕ]
2
In[]:=
P3[ϕ_]:=2^-4*ExpToTrig[((E^(I*(ϕ/2))-E^(-I*(ϕ/2)))(E^(I*(-(ϕ/2)))-E^(I*(ϕ/2))))]^2;​​FullSimplify[P3[ϕ]]
Out[]=
4
Sin
ϕ
2

In[]:=
L[ϕ_,m0_,m1_,m2_]:=((P1[ϕ])^m0)*((P2[ϕ])^m1)*((P3[ϕ])^m2)
In[]:=
L[ϕ,m0,m1,m2]
Out[]=
-m1
2
m0
4
Cos
ϕ
2

m2
4
Sin
ϕ
2

m1
(
2
Sin[ϕ]
)
In[]:=
llog[ϕ_,m1_,m2_,m3_]:=PowerExpand[Log[L[ϕ,m1,m2,m3]]]
In[]:=
llog[ϕ,m1,m2,m3]
Out[]=
-m2Log[2]+4m1LogCos
ϕ
2
+4m3LogSin
ϕ
2
+2m2Log[Sin[ϕ]]