$\text{ If }0<a<b<\frac{\pi }{b}\text{ and }f(a,b)=\frac{\tan b-\tan a}{b-a}\text{, then }$

$\text{ consider the function }f\left(x\right)=\tan x\text{, defined on }[a,b]$

$\text{ such that a, }b\in \left(0,\frac{\pi }{2}\right)$

$\begin{array}{l}{f}^{\mathrm{\prime }}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\text{ for some }c\in (a,b)\\ \Rightarrow {\sec }^{2}C=\frac{\tan b-\tan a}{b-a}\Rightarrow f(a,b)={\sec }^{2}C\\ \Rightarrow f(a,b)>1\left[\because {\sec }^{2}C>1\text{ as }\mathrm{C}\in (0,\pi /2)\right]\end{array}$