The area between $y=\tan x$ and $X-$ axis from $x=0$ to $x=\frac{\mathrm{\pi }}{4}$

The given curve is $y=\tan x$

The rough sketch of the required area is as shown below

The required area is the area of the shaded region.

The area of the shaded region is the area of the region bounded by the curve $y=\tan x$, $x=0$ and $x=\frac{\mathrm{\pi }}{4}$ with $X-$ axis

Hence the area of the shaded region is $A={\int }_0^{\frac{\mathrm{\pi }}{4}}\tan xdx$

$\begin{array}{rcl}A& =& {\int }_0^{\frac{\mathrm{\pi }}{4}}\tan xdx\\ & =& {\left[\log secx\right]}_0^{\frac{\mathrm{\pi }}{4}}\\ & =& \log \left(\sqrt{2}\right)-\log \left(1\right)\\ & =& \log \left(\sqrt{2}\right)\end{array}$

Therefore, the area of shaded region is $\boxed{\log \sqrt{2}}$