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 $cosh(x)cosh(y)$

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

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

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