We know the definition of Hyperbolic Function :
#color(green)(tanh(theta)=(e^theta-e^-theta)/(e^theta+e^-theta)#
We have to verify :
#tanh(x+y)=(tanh(x)+tanh(y))/(1+tanh(x)tanh(y)#
Let us take , LHS first.
#LHS=tanh(x+y)#
#LHS=(e^(x+y) -e^(-(x+y) ))/(e^(x+y) +e^(-(x+y) ))to(1)#
Now ,we take RHS:
#RHS=(tanh(x)+tanh(y))/(1+tanh(x)tanh(y)#
#RHS=((e^x-e^-x)/(e^x+e^-x)+(e^y-e^-y)/(e^y+e^-y))/(1+(e^x-e^-x)/(e^x+e^-x)*(e^y-e^-y)/(e^y+e^-y)#
#={(e^xe^y+color(red)(e^xe^-y)color(blue)(-e^-xe^y)-e^-xe^-y+e^xe^ycolor(red)(-e^xe^-y)color(blue)(+e^-xe^y)-e^-xe^-y)}/{(e^xe^ycolor(red)(+e^xe^-y)color(blue)(+e^-xe^y)+e^-xe^-y+e^xe^ycolor(red)(-e^xe^-y)color(blue)(-e^-xe^y)+e^-xe^-y)}#
Canceling the Red and Blue term:
#RHS=(2(e^xe^y-e^-xe^-y))/(2(e^xe^y+e^-xe^-y)#
#RHS=(e^(x+y)-e^(-(x+y)))/(e^(x+y)+e^(-(x+y)))to(2)#
From #(1) and (2)#
#RHS=LHS#
