$\tan(x) = x$. Find the values of $x$ How can I find the possible values of $x$ for:

$\tan(x)=x$

mathematically?

$$\frac{\sin(x)}{\cos(x)}=x$$
$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}...$$
$$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}...$$
Your question is equivalent to solving the equation
$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots=x-\frac{x^3}{2!}+\frac{x^5}{4!}-\cdots$$
$$x^3\left(\frac{1}{3!}-\frac{1}{2!}\right)-x^5\left(\frac{1}{5!}-\frac{1}{4!}\right)+x^7\left(\frac{1}{7!}-\frac{1}{6!}\right)+\cdots=0$$
Evidently giving
$$x=0$$
The other solutions are given by the equation
$$\left(\frac{1}{3!}-\frac{1}{2!}\right)-x^2\left(\frac{1}{5!}-\frac{1}{4!}\right)+x^4\left(\frac{1}{7!}-\frac{1}{6!}\right)-\cdots=0$$
But I don't know if there is a way to get from that to a closed form.
