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I would like to simplify the following expression :

r Cos[γ[t]] + Sqrt[l^2 - r^2/2 + 1/2 r^2 Cos[2 γ[t]]]

by using the mathematical expression :

Cos[2 x]=1 - 2 Sin[x]^2

A similar topic was already created:

How to simplify 1 - Cos[2x] to 2 Sin[x]^2?

However, I would like to use a very simple method for a Mathematica beginner.

May you advice a simple solution to obtain this kind of reduction ?

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Not sure why you would think this is a difficult problem. A simple replacement works:

r Cos[γ[t]] + Sqrt[l^2 - r^2/2 + 1/2 r^2 Cos[2 γ[t]]] /. 
  Cos[2  a_] :> 1 - 2 Sin[a]^2 // FullSimplify
(* r Cos[γ[t]] + Sqrt[l^2 - r^2 Sin[γ[t]]^2] *)
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