ГДЗ по алгебре и начала математического анализа 10 класс Алимов Задание 522

\[\boxed{\mathbf{522}\mathbf{.}}\]

\[\frac{tg\ 2a}{tg\ 4a - tg\ 2a} =\]

\[= \frac{tg\ 2a}{\frac{2\ tg\ 2a}{1 - tg^{2}\ 2a} - tg\ 2a} =\]

\[= \frac{tg\ 2a - tg^{3}\ 2a}{2\ tg\ 2a - tg\ 2a + tg^{3}\ 2a} =\]

\[= \frac{tg\ 2a - tg^{3}\ 2a}{tg\ 2a + tg^{3}\ 2a} = \frac{1 - tg^{2}\ 2a}{1 + tg^{2}\ 2a} =\]

\[= \frac{\frac{\cos^{2}{2a}}{\cos^{2}{2a}} - \frac{\sin^{2}{2a}}{\cos^{2}{2a}}}{\frac{\cos^{2}{2a}}{\cos^{2}{2a}} + \frac{\sin^{2}{2a}}{\cos^{2}{2a}}} =\]

\[= \frac{\cos^{2}{2a} - \sin^{2}{2a}}{\cos^{2}{2a} + \sin^{2}{2a}} = \frac{\cos{4a}}{1} =\]

\[= \cos{4a}\]