You can at least take a quick look at the zeros using `gnuplot` by plotting contours at zero of the equation

$$0 = \frac{2x}{x^2-1} - \tan(x)$$

<!-- language: lang-gnuplot -->

    set terminal png
    set output "test.png"
    
    set xlabel "x"
    set ylabel "y"
    
    set contour
    set cntrparam levels discrete 0
    set view map
    unset surface
    set isosamples 1000,1000
    splot 2*x/(x**2-1) - tan(x)

Since the equation is totally independent of $y$ the solutions are lines parallel to the $y$-axis.

[![enter image description here][1]][1]

To solve the equation numerically you could use GNU R with the [`rootSolve`](https://cran.r-project.org/package=rootSolve) package.

<!-- language: lang-r -->

    require("rootSolve")
    
    f <- function(x) 2*x/(x^2-1) - tan(x)
    r <- uniroot.all(f, c(-10,10))
    print(r)
    curve(f,-10,10,ylim=c(-5,5),n=1000)
    points(r,rep(0,length(r)))

Output

<!-- language: lang-none -->

     [1]  0.0000000 -9.6316827 -7.8540438 -6.5846170 -4.7124512 -3.6731884
     [7] -1.5708444 -1.3065367 -0.9999023  0.9999023  1.3065367  1.5708444
    [13]  3.6731884  4.7124512  6.5846170  7.8540438  9.6316827

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/dTCz5.png
  [2]: https://i.sstatic.net/y6d1J.png