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Question Number 126702 by Eric002 last updated on 30/Dec/20
if    tanh(x/2)=t  prove that  cosh(x)=((1+t^2 )/(1−t^2 ))
iftanhx2=tprovethatcosh(x)=1+t21t2
Answered by MJS_new last updated on 23/Dec/20
tanh α =((e^(2α) −1)/(e^(2α) +1))∧cosh α =((e^α +e^(−α) )/2)  ⇒  t=tanh (x/2) =((e^x −1)/(e^x +1))  ((1+t^2 )/(1−t^2 ))=((1+(((e^x −1)/(e^x +1)))^2 )/(1−(((e^x −1)/(e^x +1)))^2 ))=((((e^x +1)^2 +(e^x −1)^2 )/((e^x +1)^2 ))/(((e^x +1)^2 −(e^x −1)^2 )/((e^x +1)^2 )))=  =((2e^(2x) +2)/(4e^x ))=((e^x +e^(−x) )/2)=cosh x
tanhα=e2α1e2α+1coshα=eα+eα2t=tanhx2=ex1ex+11+t21t2=1+(ex1ex+1)21(ex1ex+1)2=(ex+1)2+(ex1)2(ex+1)2(ex+1)2(ex1)2(ex+1)2==2e2x+24ex=ex+ex2=coshx

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