How do you verify #tanh(x + y) = (tanh x + tanh y)/(1 + tanh x + tanh y)#?

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Answer 1 · 2018-07-31T10:43:50

Please see the explanation bellow

You need

#sinh(x+y)=sinhxcoshy+coshxsinhy#

#cosh(x+y)=coshxcoshy+sinhxsinhy#

You can either start with

#tanh(x+y)=(e^(x+y)-e^(-x-y))/(e^(x+y)+e^(-x-y))#

Or with

#tanh(x+y)=sinh(x+y)/cosh(x+y)#

#=(sinh(x)cosh(y)+sinh(y)cosh(x))/(cosh(x)cosh(y)+sinh(x)sinh(y))#

Dividing all the terms by #cosh(x)cosh(y)#

#=((sinh(x)cosh(y))/(cosh(x)cosh(y))+(sinh(y)cosh(x))/(cosh(x)cosh(y)))/((cosh(x)cosh(y))/(cosh(x)cosh(y))+(sinh(x)sinh(y))/(cosh(x)cosh(y)))#

#=(tanhx+tanhy)/(1+tanhxtanhy)#