\[\begin{array}{l} \frac{1}{{\sin {{20}^ \circ }}} - \frac{1}{{\operatorname{tg} {{40}^ \circ }}} = \frac{1}{{\sin {{20}^ \circ }}} - \frac{{\cos {{40}^ \circ }}}{{\sin {{40}^ \circ }}} = \frac{{2\cos {{20}^ \circ } - \cos {{40}^ \circ }}}{{\sin {{40}^ \circ }}} = \hfill \\ \frac{{2\cos {{20}^ \circ } - 2\cos {{60}^ \circ }\cos {{40}^ \circ }}}{{\sin {{40}^ \circ }}} = \frac{{2\cos {{20}^ \circ } - \cos {{100}^ \circ } - \cos {{20}^ \circ }}}{{\sin {{40}^ \circ }}} = \hfill \\ \frac{{\cos {{20}^ \circ } - \cos {{100}^ \circ }}}{{\sin {{40}^ \circ }}} = \frac{{2\sin {{60}^ \circ }\sin {{40}^ \circ }}}{{\sin {{40}^ \circ }}} = \sqrt 3 \hfill \\ \end{array}\]

\[\sqrt 3 \]