$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\frac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$? 
$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\dfrac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$ ?

This expression is found without mentioning the domain of $x,y,z$, but I don't think its true for all $x,y,z$ as the case with the expression for $\tan^{-1}x+\tan^{-1}y$, but I have trouble proving it.
So what is the complete expression for $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z$ ?
\begin{align}
\tan^{-1}x+\tan^{-1}y+\tan^{-1}z&=
\begin{cases}\tan^{-1}\left(\dfrac{x+y}{1-xy}\right)+\tan^{-1}z, &xy < 1 \\[1.5ex] 
\pi + \tan^{-1}\left(\dfrac{x+y}{1-xy}\right)+\tan^{-1}z, &xy>1,\:\:x,y>0 \\[1.5ex] 
-\pi + \tan^{-1}\left(\dfrac{x+y}{1-xy}\right)+\tan^{-1}z, &xy>1,\:\:x,y<0 \end{cases}\\
&=
\end{align}

Let


*

*$L(x,y,z) = \tan^{-1} x + \tan^{-1} y + \tan^{-1} z$

*$R(x,y,z) = \tan^{-1}\left(\frac{x+y+z - xyz}{1- xy - yz - zx}\right)$
By addition formula of tangent function, we have 
$$\tan L(x,y,z) = \tan R(x,y,z)$$
Since $\tan \theta$ is a periodic function with period $\pi$, there is an integer valued function $N(x,y,z)$ such that $$L(x,y,z) = R(x,y,z) + N(x,y,z)\pi$$
Since $\tan^{-1}\theta$ maps $\mathbb{R}$ into $(-\frac{\pi}{2}, \frac{\pi}{2})$, we have
$$|L(x,y,z)| < \frac{3\pi}{2} \land  |R(x,y,z)| < \frac{\pi}{2}\quad\implies\quad N(x,y,z) \in \{ 0, \pm 1 \}$$
Since $\tan^{-1} \theta$ is a continuous function for all $\theta$,
$N(x,y,z)$ will be constant over those domain where $xy+yz+zx \ne 1$.
Notice 
$$xy+yz+zx = 1 \iff 3\left(\frac{x+y+z}{\sqrt{3}}\right)^2 - ( x^2 + y^2 + z^2 ) = 2$$
is the equation of a two sheet hyperboloid centered at origin with symmetric axis pointing along the direction $(1,1,1)$. The complement of this hyperboloid consists of $3$ connected components. One can pick a point from each of these component and figure out the value of $N(x,y,z)$ over the whole component. 
The end result is
$$L(x,y,z) = R(x,y,z) + \begin{cases} 
\pi, & 1 < xy+yz+zx \land x+y+z > 0\\
0,   & 1 > xy+yz+zx\\
-\pi & 1 < xy+yz+zx \land x+y+z < 0
\end{cases}
$$
