Phương trình ${\sin ^4}x - {\sin ^4}\left( {x + \frac{\pi }{2}} \right) = 4\sin \frac{x}{2}\cos \frac{x}{2}\cos x$ có nghiệm là
A. $x = \frac{{3\pi }}{4} + k\pi $, $k \in \mathbb{Z}$.
B. $x = \frac{{3\pi }}{8} + k\frac{\pi }{2}$, $k \in \mathbb{Z}$.
C. $x = \frac{{3\pi }}{{12}} + k\pi $, $k \in \mathbb{Z}$.
D. $x = \frac{{3\pi }}{{16}} + k\frac{\pi }{2}$, $k \in \mathbb{Z}$.
A. $x = \frac{{3\pi }}{4} + k\pi $, $k \in \mathbb{Z}$.
B. $x = \frac{{3\pi }}{8} + k\frac{\pi }{2}$, $k \in \mathbb{Z}$.
C. $x = \frac{{3\pi }}{{12}} + k\pi $, $k \in \mathbb{Z}$.
D. $x = \frac{{3\pi }}{{16}} + k\frac{\pi }{2}$, $k \in \mathbb{Z}$.
Chọn B
${\sin ^4}x - {\sin ^4}\left( {x + \frac{\pi }{2}} \right) = 4\sin \frac{x}{2}\cos \frac{x}{2}\cos x$$ \Leftrightarrow {\sin ^4}x - {\cos ^4}x = 2\sin x\cos x$
$ \Leftrightarrow {\sin ^2}x - {\cos ^2}x = \sin 2x$$ \Leftrightarrow \sin 2x + \cos 2x = 0$
$ \Leftrightarrow \sqrt 2 \sin \left( {2x + \frac{\pi }{4}} \right) = 0$$ \Leftrightarrow x = - \frac{\pi }{8} + k\frac{\pi }{2}\,\,\left( {k \in \mathbb{Z}} \right)$.
${\sin ^4}x - {\sin ^4}\left( {x + \frac{\pi }{2}} \right) = 4\sin \frac{x}{2}\cos \frac{x}{2}\cos x$$ \Leftrightarrow {\sin ^4}x - {\cos ^4}x = 2\sin x\cos x$
$ \Leftrightarrow {\sin ^2}x - {\cos ^2}x = \sin 2x$$ \Leftrightarrow \sin 2x + \cos 2x = 0$
$ \Leftrightarrow \sqrt 2 \sin \left( {2x + \frac{\pi }{4}} \right) = 0$$ \Leftrightarrow x = - \frac{\pi }{8} + k\frac{\pi }{2}\,\,\left( {k \in \mathbb{Z}} \right)$.