Given:$f\left(x\right)=\tan x-x$

Differentiating on both sides, we get

$f^{\text{'}}\left(x\right)={\sec }^{2}x-1$

$\text{As,} \sec x\in (-\infty ,-1]\cup [1,\infty )$

$\therefore {\sec }^{2}x\in [1,\infty )$

$\therefore {\sec }^{2}x-1\ge 0$

$\therefore f^{\text{'}}\left(x\right)\ge 0 \text{for all} x\in R*$

$\therefore$$f\left(x\right)$ is an increasing function.

Hence, option $1$ is correct.