Sorry, Michael Rogers, for coming back so late. I was occupied with other things. Thank you very much. All of your solutions work well.

Sorry, Daniel, for coming back so late. I was occupied with other things.

Thank you very much. Your PolynomialReduce works indeed. I had pn not cleared as you assumed.

Hi Daniel, pretty strange to do that simple task with PolynomialReduce. For me at least this does not work:

In[204]:= 
PolynomialReduce[exp1, n*Sin[\[Pi]/n] - pn, {n, Sin[\[Pi]/n], pn}][[2]]

During evaluation of In[204]:= General::ivar: n Sin[\[Pi]/n] is not a valid variable.

Out[204]= 0

Performing algebraic transformations with syntactic transformations sometimes requires indirection:

2 n (r + h) Sin[\[Pi]/n] /. Sin[\[Pi]/n] -> pn/n
(*  2 pn (h + r)  *)

Usually you need to replace a simple and unique (or characteristic) factor with its equivalent.

Alternatively, you can try algebraic functions:

Eliminate[{z == 2 n (r + h) Sin[\[Pi]/n], pn == n Sin[\[Pi]/n]}, Sin[\[Pi]/n]]

(*  z == pn (2 h + 2 r)  *)

z /. First@
   Solve[{z == 2 n (r + h) Sin[\[Pi]/n], pn == n Sin[\[Pi]/n]}, 
    z, {Sin[\[Pi]/n]}] // Factor

(* 2 pn (h + r)  *)

Reduce[{z == 2 n (r + h) Sin[\[Pi]/n], pn == n Sin[\[Pi]/n]}, {z}, {Sin[\[Pi]/n], n}] // Factor

(*  (pn == 0 && z == 0) || z == 2 pn (h + r)  *)

Hello Werner,

when I start with a fresh kernel I get this:

I also did not get your point "I'm doing the calculation myself and not Wolfram": You wanted to replace an expression by another (simpler) expression and this what the code does

Hmm. I think your Replace-Rule is wrong. And it does not work. And - worst of all: By that kind of things I'm doing the calculation myself and not Wolfram. Imagine that my terms n and Sin[[Pi]/n] occur in a much larger and more complicated exp1.

exp1 /. n k_ Sin[\[Pi]/n] -> n k pn