Using addition formulas brings weird results So, I have this set of equation that I want to solve for x,y
$$
A = \tanh(x+y) \\
B = \tanh(x-y)
$$
and of course it can be solved by inverting and then summing/differencing, so I get
$$
x = \dfrac{1}{2}\bigg(arctanhA + arctanhB \bigg) \\
y = \dfrac{1}{2}\bigg(arctanhA - arctanhB \bigg)
$$
But what if I want to use addition formula (and I need to because this is just a short version of the equations that I have, where the above method cannot be used) writing
$$
A = \dfrac{\tanh x + \tanh y}{1+\tanh x \tanh y} \\
B = \dfrac{\tanh x - \tanh y}{1-\tanh x \tanh y}
$$
Now I get a quadratic equation for $\tanh x$ and I didn't figure out how to show that it gives the same result. I tried using identities and also plotting the two results but they are different. Here is the quadratic equation that I have:
$$
\tanh y = \dfrac{A - \tanh x}{1-A \tanh x} \\
B \bigg[ 1 - \tanh x \dfrac{A -\tanh x}{1- A\tanh x} \bigg] = \tanh x - \dfrac{A -\tanh x}{1- A\tanh x}
$$
and then finally for $\tanh x$
$$
(A+B) \tanh ^2 x -2 \tanh x(A+1)  + A+B=0
$$

I think your original problem with equations like
$$A = \tanh (x+y) + (1-p) \tanh x$$
is more complicated than the simplified problem.
Look at what the solutions can be (using Geogebra) for $A=0.6, B=0.3,p=0.2$ for example:

(are the equations the good ones ?).
