\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 {(x + 2x )}\cdot \cos 3x = - \sin {3x}\cdot \cos x \cdot \cos {2x}  \\
\sin {3x} \cdot \cos {3x} + \sin 3x \cdot \cos x \cdot \cos 2x  = 0\\
\sin 3x \cdot(\cos 3x + \cos x \cdot  \cos 2x)= 0 \\
\end{array}