The area between the curves y = tanx, y = cotx and X-axis in $\frac{\mathrm{\pi }}{6}\le \mathrm{x}\le \frac{\mathrm{\pi }}{3}$ is

$y=\left\{\begin{array}{l}\tan x,-\frac{\pi }{3}\le x\le \frac{\pi }{3}\\ \cot x,\frac{\pi }{6}\le x\le \frac{\pi }{2}\end{array}\right.$

$\text{ Which could be plotted as, y-axis. }$

$\begin{array}{r}\text{ Required area }={\int }_{\pi /6}^{\pi /4} (\tan x)dx+{\int }_{\pi /4}^{\pi /3} (\cot x)dx\\ =(-\log |secx|{)}_{\pi /6}^{\pi /4}+(\log \sin x{)}_{\pi /4}^{3\pi /4}\\ =-\left\{\log \frac{1}{\sqrt{2}}-\log \frac{2}{\sqrt{3}}\right\}+\left\{\log \frac{\sqrt{3}}{2}-\log \frac{1}{\sqrt{2}}\right\}\\ =2\log \frac{\sqrt{3}}{2}-2\log \frac{1}{\sqrt{2}}\\ =\log \frac{3}{4}-\log \frac{1}{2}={\log }_{\mathrm{e}}\frac{3}{2}\end{array}$