Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1073

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

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

\[= \frac{tg\ 2a \bullet \left( 1 - tg^{2}\ 2a \right)}{2\ tg\ 2a - tg\ 2a \bullet \left( 1 - tg^{2}\ 2a \right)} =\]

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

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

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

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

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

\[= \frac{\cos{4a}}{1} = \cos{4a}\]