Chỉ đúng với điều kiện A, B, C là 3 góc trong tam giác \(\Rightarrow A+B+C=\pi\)

Đặt \(\frac{A}{2}=x\) , \(\frac{B}{2}=y\); \(\frac{C}{2}=z\) \(\Rightarrow x+y+z=\frac{\pi}{2}\Rightarrow\left\{{}\begin{matrix}x+y=\frac{\pi}{2}-z\\z=\frac{\pi}{2}-\left(x+y\right)\end{matrix}\right.\)

\(cot\frac{A}{2}+cot\frac{B}{2}+cot\frac{C}{2}=cotx+coty+cotz=\frac{cosx}{sinx}+\frac{cosy}{siny}+\frac{cosz}{sinz}\)

\(=\frac{cosx.siny+cosy.sinx}{sinx.siny}+\frac{cosz}{sinz}=\frac{sin\left(x+y\right)}{sinx.siny}+\frac{cosz}{sinz}\)

\(=\frac{sin\left(\frac{\pi}{2}-z\right)}{sinx.siny}+\frac{cosz}{sinz}=\frac{cosz}{sinx.siny}+\frac{cosz}{sinz}=cosz\left(\frac{1}{sinx.siny}+\frac{1}{sinz}\right)\)

\(=\frac{cosz}{sinx.siny.sinz}\left(sinz+sinx.siny\right)=\frac{cosz}{sinx.siny.sinz}\left(sin\left(\frac{\pi}{2}-\left(x+y\right)\right)+sinxsiny\right)\)

\(=\frac{cosz}{sinx.siny.sinz}\left(cos\left(x+y\right)+sinx.siny\right)\)

\(=\frac{cosz}{sinx.siny.sinz}\left(cosx.cosy-sinx.siny+sinx.siny\right)\)

\(=\frac{cosx.cosy.cosz}{sinx.siny.sinz}=cotx.coty.cotz=cot\frac{A}{2}.cot\frac{B}{2}.cot\frac{C}{2}\)

Sửa đề: \(\cot A+\cot B+\cot C\ge\sqrt3\)

\(\cot A + \cot B = \frac{\cos A}{\sin A} + \frac{\cos B}{\sin B} = \frac{\sin(A + B)}{\sin A \sin B} = \frac{\sin C}{\sin A \sin B}\)

Ta có: \(\cot A \cot B + \cot B \cot C + \cot C \cot A = 1\)

Đặt x=cot A; y=cot B; z=cot C

Ta có: \(\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\ge0\forall x,y,z\)

=>\(x^2+y^2-2xy+y^2+z^2-2yz+x^2+z^2-2xz\ge0\)

=>\(2\left(x^2+y^2+z^2\right)\ge2\left(xy+yz+xz\right)\)

=>\(x^2+y^2+z^2\ge xy+yz+xz\)

=>\(x^2+y^2+z^2+2\left(xy+yz+xz\right)\ge xy+yz+xz+2\left(xy+yz+xz\right)\)

=>\(\left(x+y+z\right)^2\ge3\left(xy+yz+xz\right)\)

=>\((\cot A + \cot B + \cot C)^2 \ge 3(\cot A \cot B + \cot B \cot C + \cot C \cot A) = 3 \cdot 1 = 3\)

=>\(\cot A + \cot B + \cot C \ge \sqrt{3}\)

Dấu '=' xảy ra khi \(\hat{A}=\hat{B}=\hat{C}=\frac{180^0}{3}=60^0\)