$({\bf{sec}}{^2}{\bf{x}} + {\bf{cosec}}{^2}{\bf{x}} = 1{\rm{}} + {\rm{}}{\tan ^2}{\rm{x}} + {\rm{}}1{\rm{}} + {\rm{}}{\cot ^2}{\rm{x}} = {\rm{}}2{\rm{}} + {\rm{}}{\tan ^2}{\rm{x}} + {\rm{}}{\cot ^2}{\rm{x}})$

Now we have to find the minimum value of

$(\begin{array}{l} \Rightarrow {\bf{ta}}{{\bf{n}}^2}{\bf{x}} + {\cot ^2}{\bf{x}} = {\left( {\tan {\bf{x}}} \right)^2} + {\left( {\cot {\bf{x}}} \right)^2} - 2 \times \tan {\bf{x}} \times \\cot {\bf{x}} + 2 \times \tan {\bf{x}} \times \cot {\bf{x}}\\ \Rightarrow {\left( {\tan {\bf{x}} - \cot {\bf{x}}} \right)^2} + 2 = {\left( {\tan {\bf{x}} - \cot {\bf{x}}} \right)^2} + 2\end{array})$

For the value to be minimum, $(\tan {\bf{x}} - \cot {\bf{x}}\; = \;0)$