\begin{array}{lcl}
\hbox{tg}x + \hbox{tg}2x + \hbox{tg}3x = 0 \\
\hbox{tg}x + \hbox{tg}2x = -\hbox{tg}3x \\
\frac{\sin x}{\cos x} + \frac{sin {2x}}{cos {2x}} = -\frac{sin {3x}}{cos {3x}} \\
\frac{\sin x \cdot \cos {2x} + \cos x \cdot \sin {2x}}{\cos x \cdot \cos {2x}} = -\frac{sin {3x}}{cos {3x}}\\
sin {(2x + x )}\cdot \cos 3x = - \cos x \cdot \cos {2x} \cdot \sin {3x}\\
\sin {3x} \cdot \cos {3x} + \cos x * cos 2x * sin 3x = 0\\
sin 3x (cos 3x + cos x * cos 2x)= 0 \\
sin 3x [cos ( 2x + x ) + cos x * cos 2x)= 0
sin 3x [ cos x * cos 2x - sin x * sin 2x+ cos x * cos 2x)= 0\\
- sin 3x * sin x * sin 2x = 0\\
sin x \cdot sin {2x} \cdot sin {3x} = 0\\
\end{array}