If $0<\alpha <\beta <\frac{\pi }{2}$, then which of the following is not true statement. 

Consider the function  $f \left(x\right)$given by 

$f\left(x\right)=x\tan x,x\in (0,\pi /2)$

We have,  

$f^{\mathrm{\prime }}\left(x\right)=x{\sec }^{2}x+\tan x>0$for all $x\in (0,\pi /2)$

$\Rightarrow f\left(x\right)$is increasing on $(0,\pi /2)$ 

$\Rightarrow f\left(\alpha \right)<f\left(\beta \right)$ for $0<\alpha <\beta <\frac{\pi }{2}$ 

$\begin{array}{l}\Rightarrow \alpha \tan \alpha <\beta \tan \beta \\ \Rightarrow \frac{\alpha }{\beta }<\frac{\tan \beta }{\tan \alpha }\end{array}$