Gọi $x =\tan A,y= \tan B, z = \tan C$
Ta có
$\tan A. \tan B. \tan C=\tan A+ \tan B+ \tan C$
$\Leftrightarrow \tan A(\tan B. \tan C-1)= \tan B + \tan C$
$\Leftrightarrow \frac{\tan B+ \tan C}{1-\tan B. \tan C}=-tan A$
$\Leftrightarrow \tan (B+C)=\tan (\pi -A)$
$\Leftrightarrow A+B+C= \pi$
$\Leftrightarrow T = \cot A+ \cot B + \cot C \ge \sqrt 3 $