If $x+y+z=\pi ,\tan x\tan z=2$ and $\tan y\tan z=18\text{, }$ then ${\tan }^{2}z=$ 

We have 

  $\begin{array}{l}x+y+z=\pi \\ \Rightarrow \tan x+\tan y+\tan z=\tan x\tan y\tan z\\ \Rightarrow \tan x+\tan y+\tan z=2\tan y [\because \tan x\tan z=2]\\ \Rightarrow \tan x+\tan z=\tan y\\ \Rightarrow (\tan x+\tan z)\tan z=\tan y\tan z\\ \Rightarrow (\tan x+\tan z)\tan z=18\\ \Rightarrow \tan x\tan z+{\tan }^{2}z=18\\ \Rightarrow 2+{\tan }^{2}z=18\Rightarrow {\tan }^{2}z=16\end{array}$