Задача 33. Решите уравнение: \(4{\cos ^4}x-4{\cos ^2}x + 1 = 0\)
ОТВЕТ: \(\dfrac{\pi }{4} + \dfrac{{\pi k}}{2};\;\;k \in Z.\)
\(4{\cos ^4}x-4{\cos ^2}x + 1 = 0\;\;\;\; \Leftrightarrow \;\;\;\;{\left( {2{{\cos }^2}x-1} \right)^2} = 0\;\;\;\; \Leftrightarrow \;\;\;\;2{\cos ^2}x-1 = 0\;\;\;\; \Leftrightarrow \) \( \Leftrightarrow \;\;\;\;{\cos ^2}x = \dfrac{1}{2}\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{\cos x = \dfrac{{\sqrt 2 }}{2},}\\{\cos x = -\dfrac{{\sqrt 2 }}{2}}\end{array}} \right.\;\;\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \pm \dfrac{\pi }{4} + 2\pi k,}\\{x = \pm \dfrac{{3\pi }}{4} + 2\pi k}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \;\;\;\;x = \dfrac{\pi }{4} + \dfrac{{\pi k}}{2},\;\;\;\;k \in Z.\) Ответ: \(\dfrac{\pi }{4} + \dfrac{{\pi k}}{2},\,\,\,\,\,\,\,k\, \in \,Z.\)