Can I simplify an expression into form which uses my own definitions?

Daniel Lichtblau and Andrzej Koslowski posted a solution in mathgroup, which I adjusted marginally. (I like to use german identifiers, because they will never clash with Mma builtins). That's the code:

SetAttributes[termErsetzung,Listable];
termErsetzung[expr_, rep_, vars_] := 
Module[{num = Numerator[expr], den = Denominator[expr],
        hed = Head[expr], base, expon},
  If[PolynomialQ[num, vars] && PolynomialQ[den, vars] && ! NumberQ[den], 
    termErsetzung[num, rep, vars]/termErsetzung[den, rep, vars], (*else*)
    If[hed === Power && Length[expr] === 2,        
       base  = termErsetzung[expr[[1]], rep, vars];
       expon = termErsetzung[expr[[2]], rep, vars];
       PolynomialReduce[base^expon, rep, vars][[2]],        (*else*)
      If[Head[Evaluate[hed]] === Symbol && 
        MemberQ[Attributes[Evaluate[hed]], NumericFunction], 
        Map[termErsetzung[#, rep, vars] &, expr],    (*else*)
       PolynomialReduce[expr, rep, vars][[2]] ]]]
];

TermErsetzung[rep_Equal,vars_][expr_]:=
  termErsetzung[expr,Evaluate[Subtract@@rep],vars]//Union;

Usage is like this:

a*b/(a + a*Cos[a/b]) // TermErsetzung[k b == a, b]

a/(k (1 + Cos[k]))

The first parameter is the "replacement equation", the second the variable (or list of variables) to be eliminated:

a*b/(a + a*Cos[a/b]) // TermErsetzung[k b == a, {a, b}] 

{b/(1 + Cos[k]), a/(k (1 + Cos[k]))}

This general question has been raised on the main StackOverflow site. The answers to these questions may be helpful:

How to reduce the number of independent variables in mathematica

Question on “smart” replacing in mathematica

How to perform a complicated change of variables for a polynomial (in Mathematica)

Get mathematica to simplify expression with another equation

Look specifically at the answers by Daniel Lichtblau.

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More questions on the topic here on Mathematica.SE:

Can I simplify an expression into form which uses my own definitions? (this question)

How do I replace a variable in a polynomial?

Replacing composite variables by a single variable

Replace expressions with symbols

Reduce an equation by putting a new variable

Find subexpression to minimize leafcount after replacment with temporary variable

Common subexpression from two expressions

Rewriting one expression using a variable representing another expression

Replacing a sum of expressions

In this simple example you can just use a rule. In more complex cases it might not be straighforward to generalize.

Simplify[a*b/(a + a*Cos[a/b]) /. a -> k b]

(* b/(1 + Cos[k]) *)