Show that there is a number x ∈ [ π / 2 , π ]...

Yes, you can. Consider $f(x)=\tan x+x$. Notice that $f(\pi /2+ϵ)\approx -\mathrm{\infty }$ for small $ϵ$ (more formally, the right limit as $x\to \pi /2$ of $f(x)$ is $-\mathrm{\infty }$) while $f(\pi )=\pi$. Now the intermediate value theorem gives you want you want.

Your problem is equivalent to proving that
$\begin{array}{}\text{(1)}& f(z)=z\cot z=z^2+\frac{\pi z}{2}=g(z)\end{array}$
for some $z\in I=(0,\frac{\pi }{2})$. That is trivial since $f(z)$ decreases from 1 to 0 on $I$ while $g(z)$ increases from 0 to $\frac{{\pi }^2}{2}$ on $I$, and both $f(z)$ and $g(z)$ are continuous over $I$. By approximating $z\cot z$ with $1-\frac{z^2}{3}$ we also get:
$z\approx \frac{-3\pi +\sqrt{192+9{\pi }^2}}{16}$
hence the solution to the original problem is about:
$\begin{array}{}\text{(2)}& x\approx \frac{5\pi +\sqrt{192+9{\pi }^2}}{16}.\end{array}$