Assertion (A) : In a $∆\mathrm{ABC}$ the minimum value of ${\tan }^2 \frac{\mathrm{A}}{2} + {\tan }^2\frac{B}{2} + {\tan }^2\frac{C}{2}$ is 1

Reason (R) : ${\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2 - \mathrm{ab} - \mathrm{bc} -\mathrm{ca} \ge 0$

A:

$$\begin{gathered} \mathrm{A}+\mathrm{B}+\mathrm{C}=\mathrm{\pi } \Rightarrow \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\frac{\mathrm{C}}{2}=\frac{\mathrm{\pi }}{2} \\ \tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=1 \\ {\left(\tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2+{\left(\tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2+{\left(\tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2\ge 0 \\ \Rightarrow {\tan }^2 \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\tan }^2 \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\tan }^2 \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\left(\tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\tan \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2+\tan \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \tan \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\right.\right)\ge 0 \\ {\tan }^2 \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\tan }^2 \raisebox{1ex}{$\mathrm{b}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\tan }^2 \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-1\ge 0 \\ {\tan }^2 \raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\tan }^2 \raisebox{1ex}{$\mathrm{B}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. +{\tan }^2 \raisebox{1ex}{$\mathrm{C}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\ge 1 \end{gathered}$$

R:

$\begin{array}{rcl} & & {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\\ & =& \frac{1}{2}\left(2{\mathrm{a}}^2+2{\mathrm{b}}^2+2{\mathrm{c}}^2-2\mathrm{ab}-2\mathrm{bc}-2\mathrm{ca}\right)\\ & =& \frac{1}{2}\left[{\left(\mathrm{a}-\mathrm{b}\right)}^2+(\mathrm{b}-\mathrm{c}0+{(\mathrm{c}-\mathrm{a})}^2\right]\\ & \ge & 0\end{array}$