For the equation $2\mathrm{x}=\tan \left(2{\tan }^{-1}\mathrm{a}\right)+2\tan \left({\tan }^{-1}\mathrm{a}+{\tan }^{-1}{\mathrm{a}}^3\right),$ which of the following is valid ?

$$\begin{gathered} 2\mathrm{x}=\tan \left(2{\tan }^{-1}\mathrm{a}\right)+2\tan \left({\tan }^{-1}\mathrm{a}+{\tan }^{-1}{\mathrm{a}}^3\right) \\ 2x=\tan ({\tan }^{-1}\left(\frac{2a}{1-a^2}\right)+2\tan \left({\tan }^{-1}\left(\frac{a+a^3}{1-a^4}\right)\right) \\ \Rightarrow 2\mathrm{x}=\frac{2\mathrm{a}}{1-{\mathrm{a}}^2}+\frac{2\left(\mathrm{a}+{\mathrm{a}}^3\right)}{1-{\mathrm{a}}^4} \\ \Rightarrow 2x=\frac{2\mathrm{a}}{1-{\mathrm{a}}^2}+\frac{2a}{1-a^2}=\frac{4\mathrm{a}}{1-{\mathrm{a}}^2} \\ \Rightarrow x\left(1-{\mathrm{a}}^2\right)=2a \\ \therefore {\mathrm{a}}^2\mathrm{x}+2\mathrm{a}=\mathrm{x} \end{gathered}$$