If you want the solution in cartesian coordinates, which now appears what you want, you can use:

TransformedField["Polar" -> "Cartesian", r == Sqrt[Cos[\[CapitalTheta]]^2 + 1], {r, \[Theta]} -> {x, y}]

(*Sqrt[x^2 + y^2] == Sqrt[1 + Cos[\[CapitalTheta]]^2]*)

That still contains the $Theta$ so you can use

TransformedField["Polar" -> "Cartesian", r == Sqrt[Cos[\[CapitalTheta]]^2 + 1], {r, \[Theta]} -> {x, y}] /. \[CapitalTheta] -> 
  CoordinateTransform["Cartesian" -> "Polar", {x, y}][[2]]//FullSimplify

(*x^2 + y^2 == 1 + x^2/(x^2 + y^2)*)

To convert that to the equation you desire, you can multiply both sides by $(x^2+y^2)$:

Map[#*(x^2 + y^2) &, %] // FullSimplify

(* (x^2 + y^2)^2 == 2 x^2 + y^2*)

Cheers,

M.