If $a \tan \mathrm{\alpha }+\sqrt{\left({\mathrm{a}}^2-1\right)}\tan \mathrm{\beta }+\sqrt{\left({\mathrm{a}}^2+1\right)}\tan \mathrm{\gamma }=2\mathrm{a}$ where a is constant and  $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are variable angles. Then the least value of 2727 $\left({\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\right)$ must be

we have , ${\left(\mathrm{atan}\mathrm{\beta }-\sqrt{\left({\mathrm{a}}^2-1\right)}\tan \mathrm{\alpha }\right)}^2+\left(\sqrt{\left({\mathrm{a}}^2-1\right)}\tan \mathrm{\gamma }\right.{\left.-\sqrt{\left({\mathrm{a}}^2+1\right)}\tan \mathrm{\beta }\right)}^2+{\left(\sqrt{\left({\mathrm{a}}^2+1\right)}\tan \mathrm{\alpha }-\mathrm{atan}\mathrm{\gamma }\right)}^2\ge 0$

$\Rightarrow \left({\mathrm{a}}^2+{\mathrm{a}}^2-1+{\mathrm{a}}^2+1\right)\left({\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\right)-{\left\{\mathrm{atan}\mathrm{\alpha }+\sqrt{\left({\mathrm{a}}^2-1\right)}\tan \mathrm{\beta }+\sqrt{\left({\mathrm{a}}^2+1\right)}\tan \mathrm{\gamma }\right\}}^2\ge 0$   [using Lagrange's identity]

$\begin{array}{l}\Rightarrow 3{\mathrm{a}}^2\left({\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\right)-(2\mathrm{a}{)}^2\ge 0\\ \therefore 3\left({\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\right)\ge 4\end{array}$

Hence $2727\left({\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\right)\ge 3636$

Least value is 3636.