If $\Delta ABC,\tan A+\tan B+\tan C=6$ and $\tan A\tan B=2$ then ${\sin }^{2}A:{\sin }^{2}B:{\sin }^{2}C$

$\Rightarrow \tan A+\tan B+\tan C=\tan A\tan B\tan C$

$\Rightarrow 6=2\tan C$ $\left(\because \text{ Given }\tan A\tan B=2\text{and }\tan A+\tan B+\tan C=6\right)$

$\Rightarrow \tan C=3$

$\Rightarrow \tan A+\tan B=3$

$\Rightarrow \tan A\tan B=2$

Let ${\left(\tan A-\tan B\right)}^2={\left(\tan A+\tan B\right)}^2-4\tan A.\tan B$

$=9-4\left(2\right)$

$=1$

               $\tan A=2$

$\tan A=\frac{2k}{k},\tan B=\frac{k}{k},\tan C=\frac{3k}{k}$

$\sin A=\frac{2k}{\sqrt{5}k}=\frac{2}{\sqrt{5}},\sin B=\frac{k}{\sqrt{2}k}=\frac{1}{\sqrt{2}},\sin C=\frac{3}{\sqrt{10}}$

$\Rightarrow {\sin }^{2}A:{\sin }^{2}B:{\sin }^{2}C=\frac{4}{5}:\frac{1}{2}:\frac{9}{10}$

$=\frac{4\times 2}{5\times 2}:\frac{1\times 5}{2\times 5}:\frac{9}{10}$

$=\frac{8}{10}:\frac{5}{10}:\frac{9}{10}$

$=8:5:9$

$\tan A=1,\tan B=2,\tan C=3$

Similarly  ${\sin }^{2}A:{\sin }^{2}B:{\sin }^{2}C=5:8:9$