Giải phương trình: \(4\left( {{{\cos }^4}x + {{\si...

D

- Hướng dẫn giải

Phương pháp giải:

Giải chi tiết:

\(\begin{array}{l}{\rm{     }}4\left( {{{\cos }^4}x + {{\sin }^4}x} \right) + \sqrt 3 \sin 4x = 2\\ \Leftrightarrow 4\left( {co{s^4}x + 2si{n^2}x.co{s^2}x + si{n^4}x} \right) - 8si{n^2}x.{\cos ^2}x + \sqrt 3 \sin 4x = 2\\ \Leftrightarrow 4{\left( {co{s^2}x + si{n^2}x} \right)^2} - 2{\left( {2.sinxcosx} \right)^2} + \sqrt 3 \sin 4x = 2\\ \Leftrightarrow 4 - 2{\sin ^2}2x + \sqrt 3 .\sin 4x = 2\\ \Leftrightarrow 4 - 1 + \cos 4x + \sqrt 3 \sin 4x = 2\\ \Leftrightarrow \cos 4x + \sqrt 3 .\sin 4x =  - 1\\ \Leftrightarrow \frac{1}{2}\cos 4x + \frac{{\sqrt 3 }}{2}\sin 4x =  - \frac{1}{2}\\ \Leftrightarrow \sin \frac{\pi }{6}.\cos 4x + \cos \frac{\pi }{6}.\sin 4x =  - \frac{1}{2}\\ \Leftrightarrow \sin \left( {\frac{\pi }{6} + 4x} \right) =  - \frac{1}{2} = \sin \frac{{ - \pi }}{6}\\ \Leftrightarrow \left[ \begin{array}{l}\frac{\pi }{6} + 4x =  - \frac{\pi }{6} + k2\pi \\\frac{\pi }{6} + 4x = \frac{{7\pi }}{6} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{{12}} + \frac{{k\pi }}{2}\\x = \frac{\pi }{4} + \frac{{k\pi }}{2}\end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\end{array}\)

Chọn D.