How many solutions does the equation tan(x)=x have on the interval ]0,2π[ (note that this is an open interval).∞.2.1.

The equation tan(x) = x does not have a general algebraic solution, so it must be solved graphically or numerically.

Here are the steps to solve it graphically:

1. Draw the graph of y = tan(x) on the interval ]0, 2π[. The graph will have vertical asymptotes at x = π/2 and x = 3π/2, and it will pass through the origin.

2. Draw the graph of y = x on the same set of axes. This is a straight line passing through the origin.

3. The solutions to the equation tan(x) = x are the x-coordinates of the points where these two graphs intersect.

By observing the graphs, we can see that there is one intersection point in the interval ]0, 2π[, so the equation tan(x) = x has one solution in this interval.

Please note that this is a rough estimation. For a precise numerical solution, you would need to use a numerical method such as the bisection method, Newton's method, or a calculator or software capable of finding such solutions.