Solve for $\displaystyle \mathrm{x}$, \[2 \tan ^{-1} \mathrm{x}+\sin ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)=4 \sqrt{3} \]

\[2 \tan ^{-1} \mathrm{x}+\sin ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)=4 \sqrt{3} \Rightarrow 2 \tan ^{-1} \mathrm{x}+2 \tan ^{-1} \mathrm{x}=4 \sqrt{3} \] $\displaystyle \Rightarrow \boldsymbol{\operatorname { t a n }}^{-\mathbf{1}} \mathrm{x}=\sqrt{3} \notin\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ $\displaystyle \therefore \mathrm{x}=\boldsymbol{\operatorname { t a n }}(\sqrt{3})$ which has no solution.