If in the expansion of ${\left(\frac{1}{\mathrm{x}}+\mathrm{x} \tan \mathrm{x}\right)}^2$, the ratio of 4^th term to its 2^nd term  is $\frac{2}{27}{\mathrm{\pi }}^2$, then the value of x can be

We have, ${\mathrm{T}}_4{=}^5{\mathrm{C}}_3{\left(\frac{1}{\mathrm{x}}\right)}^{5-3}(\mathrm{xtan}\mathrm{x}{)}^3=10{\mathrm{xtan}}^3\mathrm{x}$ and ${\mathrm{T}}_2{=}^5{\mathrm{C}}_1{\left(\frac{1}{\mathrm{x}}\right)}^{5-1}(\mathrm{xtan}\mathrm{x})=\frac{5\tan \mathrm{x}}{{\mathrm{x}}^3}$

we are given, $\frac{{\mathrm{T}}_4}{{\mathrm{T}}_2}=\frac{2}{27}{\mathrm{\pi }}^4\Rightarrow 2{\mathrm{x}}^4{\tan }^2\mathrm{x}=\frac{2}{27}{\mathrm{\pi }}^4$

$\Rightarrow {\mathrm{x}}^2\tan \mathrm{x}=\pm \frac{1}{3\sqrt{3}}{\mathrm{\pi }}^2$

it is possible, when $\mathrm{x}=\pm \mathrm{\pi }/3$