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help on Rewrite rules


I am having trouble simplifying expressions in mathematica. My epxerissions 
involve two parameters x & y that parameterise a unit circle so that

x^2 + y^2 =1

Without chosing a particular parameterization for x and y I want to simplify 
epxpertions of the form:


3x^2 + 2y^2               -----> 2 + x^2
or
(x^2 + y^2)^(1/2)        -----> 1

and more importantly for should perform the simplification

A[x^2] + B [y^2] + C[x^2+y^2]    -----> A[ x^2] + B[ y^2] + C[1]

where A,B&C are functions.

I assume that this can me achieved with a relatively simple rewrite rule but 
I have had very limited success making this work. Can anyone suggest a 
solution,

Many thanks
Jon Palmer

P.S. I also want to expand the problem to that of three variables x,y&z 
parameterizing a unit sphere but I suspect that this will be obvious form 
the solution of the unit circle problem.

 


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