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

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

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.

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

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

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.

To solve the equation numerically you could use GNU R with the rootSolve package.

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

 [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