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I talked to a GIAC developer. The command to use is eliminate:
eliminate([x - A*cost*cos2t + sint*sin2t, y - A*sint*cos2t - cost*sin2t, z - B*cos2t, A^2 + B^2 - 1, cost^2 + sint^2 - 1, cos2t - cost^2 + sint^2, sin2t - 2*sint*cost], [cost, sint, cos2t, sin2t])
The result is immediate.
He confirmed we should avoid the plex ordering.
