Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1000

\[\boxed{\mathbf{1000.}}\]

\[\sin a + \cos a = p\]

\[p^{2} = \left( \sin a + \cos a \right)^{2} = \sin^{2}a +\]

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

\[= 1 + 2\sin a \cdot \cos a;\]

\[\sin a \cdot \cos a = \frac{1}{2}\left( p^{2} - 1 \right)\]

\[1)\sin a - \cos a\]

\[\left( \sin a - \cos a \right)^{2} = \sin^{2}a -\]

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

\[= 1 - \left( p^{2} - 1 \right) = 2 - p^{2}.\]

\[\sin a - \cos a = \pm \sqrt{2 - p^{2}}.\]

\[2)\ \left( \sin a + \cos a \right)^{2} = p^{2}\]

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

\[1 + \sin{2a} = p^{2}\]

\[\sin{2a} = p^{2} - 1\]

\[\sin^{4}a + \cos^{4}a =\]

\[= \left( \sin a + \cos a \right)^{2} -\]

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

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

\[= 1 - \frac{1}{2}\left( p^{2} - 1 \right)^{2} = 1 -\]

\[- \frac{1}{2} \cdot \left( p^{4} - 2p^{2} + 1 \right) =\]

\[= 1 - \frac{1}{2}p^{4} + p^{2} - \frac{1}{2} = \ \]

\[= - \frac{1}{2}p^{4} + p^{2} + \frac{1}{2}\text{.\ }\]