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

1 Answer
Jul 31, 2018

Please see the explanation bellow

Explanation:

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)