$\text{ Let }\mathrm{S}=\left\{\mathrm{x}\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right):9^{1-{\tan }^2\mathrm{x}}+9^{{\tan }^2\mathrm{x}}=10\right\}\text{ and }\mathrm{\beta }=\sum _{\mathrm{x}\in \mathrm{s}} {\tan }^2\left(\frac{\mathrm{x}}{3}\right)\text{, then }\frac{1}{6}(\mathrm{\beta }-14{)}^2\text{ is }$equal to

$\begin{array}{l}{9}^{1-{\tan }^2\mathrm{x}}+9^{{\tan }^2\mathrm{x}}=10\text{ let }{9}^{{\tan }^2\mathrm{x}}=\mathrm{t}\\ \Rightarrow \frac{9}{\mathrm{t}}+\mathrm{t}=10\Rightarrow 9+{\mathrm{t}}^2=10\mathrm{t}\\ \Rightarrow {\mathrm{t}}^2-10\mathrm{t}+9=0\Rightarrow (\mathrm{t}-1)(\mathrm{t}-9)=0\\ \mathrm{t}=1\text{ or }\mathrm{t}=9\\ 9^{{\tan }^2\mathrm{x}}=1\text{ or }{9}^{{\tan }^2\mathrm{x}}=9\\ \Rightarrow {\tan }^2\mathrm{x}=0\text{ or }{\tan }^2\mathrm{x}=1\\ \Rightarrow \tan \mathrm{x}=0\text{ or }\tan \mathrm{x}=\pm 1\\ \mathrm{x}=0,\frac{\mathrm{\pi }}{4},-\frac{\mathrm{\pi }}{4}\\ \Rightarrow \sum {\tan }^2\left(\frac{\mathrm{x}}{3}\right)={\tan }^{2}0+{\tan }^2\left(\frac{\mathrm{\pi }}{12}\right)+{\tan }^2\left(\frac{-\mathrm{\pi }}{12}\right)\\ =0+2{\tan }^2\left(\frac{\mathrm{\pi }}{12}\right)=2{\tan }^2{15}^0=2(2-\sqrt{3}{)}^2=14-8\sqrt{3}\\ \Rightarrow \frac{1}{6}(\mathrm{\beta }-14{)}^2=\frac{1}{6}(-8\sqrt{3}{)}^2=\frac{8\times 8\times 3}{6}=32\end{array}$