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We report on the comparison between temperature-dependent magneto-absorption and magnetotransport spectroscopy of HgTe/CdHgTe quantum wells in terms of the detection of the phase transition between the topological insulator and band insulator states. Our results demonstrate that temperature-dependent magnetospectroscopy is a powerful tool to discriminate trivial and topological insulator phases, yet the magnetotransport method is shown to have advantages for the clear manifestation of the phase transition with accurate quantitative values of the transition parameter (i.e., critical magnetic field Bc).