Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1085

\[\boxed{\mathbf{1085}\mathbf{.}}\]

\[1)\sin\left( \frac{\pi}{4} + a \right) - \cos\left( \frac{\pi}{4} - a \right) = 0\]

\[\sin\left( \frac{\pi}{2} - \left( \frac{\pi}{4} - a \right) \right) -\]

\[- \cos\left( \frac{\pi}{4} - a \right) = 0\]

\[\cos\left( \frac{\pi}{4} - a \right) - \cos\left( \frac{\pi}{4} - a \right) = 0\]

\[0 = 0\]

\[Что\ и\ требовалось\ доказать.\]

\[2)\cos\left( \frac{\pi}{6} - a \right) - \sin\left( \frac{\pi}{3} + a \right) = 0\]

\[\cos\left( \frac{\pi}{6} - a \right) -\]

\[- \sin\left( \frac{\pi}{2} - \left( \frac{\pi}{6} - a \right) \right) = 0\]

\[\cos\left( \frac{\pi}{6} - a \right) - \cos\left( \frac{\pi}{6} - a \right) = 0\]

\[0 = 0\]

\[Что\ и\ требовалось\ доказать.\]

\[3)\ \frac{\sin\left( \frac{3\pi}{2} - a \right)}{\text{tg}(\pi + a)} \bullet \frac{\text{ctg}\left( \frac{\pi}{2} + a \right)}{\text{tg}\left( a - \frac{3\pi}{2} \right)} =\]

\[= - \sin a\]

\[\frac{- \cos a}{\text{tg\ a}} \bullet \frac{- tg\ a}{\text{tg}\left( \pi - \left( \frac{\pi}{2} + a \right) \right)} =\]

\[= - \sin a\]

\[\frac{\cos a}{1} \bullet \frac{1}{\text{tg}\left( \frac{\pi}{2} + a \right)} = - \sin a\]

\[\cos a \bullet \frac{1}{- ctg\ a} = - \sin a\]

\[\cos a \bullet \left( - \frac{\sin a}{\cos a} \right) = - \sin a\]

\[- \sin a = - \sin a\]

\[Что\ и\ требовалось\ доказать.\]