Simplifying a power of complex exponential quotient

Question

I have an expression of the form $(\frac{-1+e^{i\,N\,\phi}}{-1+e^{i \phi}})^n$.
I want to use mma to hopefully simplify it.

I have tried ComplexExpand[], PowerExpand[] with simplifying assumption in Refine[] that $N,\phi$ are real. But on using Simplify[] or FullSimplify[]. Either mma does nothing or gives a complicated answer.

a = PowerExpand[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(N - 1\)]
\*SuperscriptBox[\(E\), \((I\ k\ \[Phi])\)]\)]

FullSimplify[ComplexExpand[a^n]]

Refine[FullSimplify[PowerExpand[a^n]], 
 Assumptions -> {N \[Element] Reals, \[Phi] \[Element] Reals}]
```

Answer 1

Maybe:

 Func = Simplify[((-1 + E^(I M \[Phi]))/(-1 + E^(I \[Phi])) // 
    ExpToTrig // TrigFactor // FullSimplify)^n, 
 Assumptions -> {M \[Element] Reals, \[Phi] \[Element] Reals, 
 n > 0}] // PowerExpand
 
 ComplexExpand[Re[Func[[1]]], TargetFunctions -> {Re, Im}]*Func[[2 ;; 3]](*Real part*)

 (*Cos[1/2 (-1 + M) n \[Phi]] Csc[\[Phi]/2]^n Sin[(M \[Phi])/2]^n*)

$$\left(\sum _{k=0}^{M-1} e^{i k \phi }\right){}^n=\cos \left(\frac{1}{2} (-1+M) n \phi \right) \csc ^n\left(\frac{\phi }{2}\right) \sin ^n\left(\frac{M \phi }{2}\right)$$