Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1001

\[\boxed{\mathbf{1001.}}\]

\[tg\ a + ctg\ a = m\]

\[1)\ tg^{2}a + ctg^{2}a = m^{2} - 2\]

\[(tg\ a + ctg\ a)^{2} = m^{2}\]

\[tg^{2}a + 2tg\ a\ ctg\ a +\]

\[+ ctg^{2}a = m^{2}\]

\[tg^{2}\ a + 2 \cdot 1 + ctg^{2}a = m^{2}\]

\[tg^{2}a + ctg^{2}a = m^{2} - 2.\]

\[2)\ tg\ a - ctg\ a = \pm \sqrt{m^{2} - 4}\]

\[(tg\ a - ctg\ a)^{2} = tg^{2}a -\]

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

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

\[= m^{2} - 4.\]

\[3)\ tg^{3}a + ctg^{3}a = m^{3} - 3m\ \]

\[(tg\ a + ctg\ a)^{3} = m^{3}\]

\[tg^{3}a + ctg^{3}a + 3tg^{2}a\ ctg\ a +\]

\[+ 3ctg^{2}a\ tga = m^{3}\]

\[tg^{3}a + ctg^{3}a +\]

\[+ 3(tg\ a + ctg\ a) = m^{3}\]

\[tg^{3}a + ctg^{3}a = m^{3} - 3m.\]