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We propose a method for construction of exactly solvable ring-shaped potentials where the linear homogeneous second-order differential equation satisfied by special function is subjected to the extended transformation comprising a coordinate transformation and a functional transformation to retrieve the standard Schr?dinger polar angle equation form in non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corresponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomials and the wave functions for the constructed ring-shaped potentials are normalized.