Giải các phương trình sau
1) $2\tan3x - 3\tan2x= \tan^22x.\tan3x$                    (1)
2) $2\cot 2x - 3\cot 3x= \tan2x$                  (2)
$1)$ Điều kiện : $c{\rm{os}}3{\rm{x}} \ne 0,c{\rm{os2x}} \ne 0$
    $\left( 1 \right) \Leftrightarrow 2\left( {tg3{\rm{x}} - tg2{\rm{x}}} \right) = tg2{\rm{x}}\left( {1 + tg2{\rm{x}}.tg3{\rm{x}}} \right)$        $(3)$
Trường hợp : $1 + tg2{\rm{x}}{\rm{.}}tg3{\rm{x  = 0}}$ (4) sẽ không thỏa mãn vì nếu thỏa mãn sẽ suy ra $tg3{\rm{x}} = tg2{\rm{x}} \Rightarrow {\rm{1 + t}}{{\rm{g}}^2}2{\rm{x}} = 0$ : vô lý
Trường hợp : $1 + tg2{\rm{x}}{\rm{.}}tg3{\rm{x }} \ne {\rm{ 0}}$$ \Leftrightarrow c{\rm{osx }} \ne {\rm{ }}0$
(4) $ \Leftrightarrow 2.\frac{{tg3{\rm{x}} - tg2{\rm{x}}}}{{1 + tg2{\rm{x}}.tg3{\rm{x}}}} = tg2{\rm{x}} \Leftrightarrow 2tg{\rm{x}} = tg2{\rm{x}}$

Kết quả : $tg{\rm{x}} = 0 \Leftrightarrow x = k\pi $ ( thỏa mãn các điều kiện $c{\rm{os}}3{\rm{x}} \ne 0,c{\rm{os2x}} \ne 0,c{\rm{osx}} \ne 0$)

$2)$ Điều kiện : $\sin 2{\rm{x}} \ne 0$, $c{\rm{os}}2{\rm{x}} \ne 0$, $\sin 3{\rm{x}} \ne 0$
$\left( 2 \right) \Leftrightarrow 2\left( {\cot g2{\rm{x}} - \cot g3{\rm{x}}} \right) = tg2{\rm{x}} + \cot g3{\rm{x}}$
       $ \Leftrightarrow \frac{{2{\mathop{\rm s}\nolimits} {\rm{inx}}}}{{\sin 2{\rm{x}}.\sin 3{\rm{x}}}} = \frac{{c{\rm{osx}}}}{{c{\rm{os}}2{\rm{x}}.\sin 3{\rm{x}}}}$
       $ \Leftrightarrow 2\sin {\rm{x}}c{\rm{os}}2{\rm{x}} = \cos {\rm{x}}\sin 2{\rm{x}} \Leftrightarrow {\mathop{\rm s}\nolimits} {\rm{inx}}\left( {c{\rm{os}}2{\rm{x}} - 1} \right) = 0$
Kết quả : $\left[ \begin{array}{l}
{\mathop{\rm s}\nolimits} {\rm{inx}} = 0      {\rm{(loại)  vì   làm  cho  sin2x = 0}}\\
c{\rm{os}}2{\rm{x}} = 1     {\rm{  (loại)  vì  sin2x = 0}}
\end{array} \right.$
Vậy phương trình đã cho vô nghiệm.

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