Approximate value of hyperbolic tangent in certain case

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I am reading Thé Nature of Magnetism. While reading, I came across a particular approximation of hyperbolic tangent

  • while in first case $T>T_c$ , it is just Taylor series,
  • in case $T < T_c$ ( where an additional term is added to the argument of the hyperbolic tangent) , the approximation is not clear for me.

Snapshot of Pages

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Any hints? Please Thank you very much for your help