\[\operatorname{tg} 11x = \frac{{\operatorname{tg} x\left( {11 - 165{{\operatorname{tg} }^2}x + 462{{\operatorname{tg} }^4}x - 330{{\operatorname{tg} }^6}x + 55{{\operatorname{tg} }^8}x - {{\operatorname{tg} }^{10}}x} \right)}}{{1 - 55{{\operatorname{tg} }^2}x + 330{{\operatorname{tg} }^4}x - 462{{\operatorname{tg} }^6}x + 165{{\operatorname{tg} }^8}x - 11{{\operatorname{tg} }^{10}}x}}\]

\[{x_1} = - {\operatorname{tg} ^2}\frac{\pi }{{11}},{\text{ }}{x_2} = - {\operatorname{tg} ^2}\frac{{2\pi }}{{11}},{\text{ }}{x_3} = - {\operatorname{tg} ^2}\frac{{3\pi }}{{11}},{\text{ }}{x_4} = - {\operatorname{tg} ^2}\frac{{4\pi }}{{11}},{\text{ }}{x_5} = - {\operatorname{tg} ^2}\frac{{5\pi }}{{11}}\]