'Approximate value of hyperpolic tangent in certain case

Reading this interesting Book ( thé nature of magnetism) , I came across a particular approximation of hyperpolic tangent, while in first case T bigger than Tc , it is just Taylor series, in case T smaller than Tc ( where an additional terme is added to the argument of the hyperbolic tangent) , the approximation is not clear for me . I attached a screenshot of the pages.

Any hints ? Please Thank you very much for your help Bestenter image description here For more context see https://archive.org/details/TheNatureOfMagnetismKaganov/page/n76/mode/2up?view=theater



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