$1)$ CMR điều kiện cần và đủ để $\triangle ABC$ đề là$\frac{1}{sinA}+\frac{1}{sinB}+\frac{1}{sinC}-(cotA+cotB+cotC)=\sqrt{3}$

$2)$ cho $x,y,z$ là các số thực dương thỏa mãn $x^2+y^2+z^2\leqslant 3$

tìm min của $P=\frac{1}{1+xy}+\frac{1}{1+yz}+\frac{1}{1+zx}$
$\begin{array}{l}
1)\,\frac{1}{{\sin A}} + \frac{1}{{\sin B}} + \frac{1}{{\sin C}} - (\cot A + \cot B + \cot C) =
\sqrt 3 \\
 \Leftrightarrow \frac{{1 - \cos A}}{{\sin A}} + \frac{{1 - \cos B}}{{\sin B}} + \frac{{1 - \cos
C}}{{\sin C}} = \sqrt 3 \\
 \Leftrightarrow \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2} = \sqrt 3
\end{array}$
$ \Leftrightarrow {\tan ^2}\frac{A}{2} + {\tan ^2}\frac{B}{2} + {\tan ^2}\frac{C}{2} = 1$

$ \Leftrightarrow \tan \frac{A}{2}\tan \frac{B}{2} + \tan \frac{B}{2}\tan \frac{C}{2} + \tan
\frac{A}{2}\tan \frac{C}{2} = 1  (\forall \Delta ABC)$
$\begin{array}{l}
 \Leftrightarrow {\tan ^2}\frac{A}{2} + {\tan ^2}\frac{B}{2} + {\tan ^2}\frac{C}{2} = \tan
\frac{A}{2}\tan \frac{B}{2} + \tan \frac{B}{2}\tan \frac{C}{2} + \tan \frac{A}{2}\tan
\frac{C}{2}\\
 \Leftrightarrow \frac{1}{2}\left[ {{{\left( {\tan \frac{A}{2} - \tan \frac{B}{2}} \right)}^2} +
{{\left( {\tan \frac{B}{2} - \tan \frac{C}{2}} \right)}^2} + {{\left( {\tan \frac{A}{2} - \tan
\frac{C}{2}} \right)}^2}} \right] = 0\\
 \Leftrightarrow \tan \frac{A}{2} = \tan \frac{B}{2} = \tan \frac{C}{2}
\end{array}$
$ \Leftrightarrow \frac{A}{2} = \frac{B}{2} = \frac{C}{2} \Leftrightarrow \Delta ABC$ đều.

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