Nghiệm của phương trình

A

- Hướng dẫn giải

Chọn A

Ta có:  $- {\sin }^{3}x + {\cos }^{3}x = \sin x –\cos x$

$\begin{array}{l}\iff (c\text{os x- sin x)}.(c{\text{os}}^{2}x+c{\text{osx. sin x+sin}}^{2}x)+\text{\hspace{0.17em}}(c\text{os x - sin x) = 0}\\ \iff (c\text{os x- sin x)}.(1+c\text{osx. sin x})+\text{\hspace{0.17em}}(c\text{os x - sin x) = 0}\\ \iff (\text{}c\text{osx - sin x ). (1+}c\text{os x. sinx}+\text{}1)=0\\ \iff (\text{}c\text{osx - sin x ). (2+\hspace{0.17em}\hspace{0.17em}}\frac{\sin 2x}{2}\text{ )}=0\iff \left[\begin{array}{c}\text{cosx - sinx}=0\\ \frac{\sin 2x}{2}=-2\end{array}\right.\end{array}$

$\begin{array}{l}\text{cosx - sin x =0 }\iff \sqrt{2}\text{cos }\left(x+\text{\hspace{0.17em}\hspace{0.17em}}\frac{\pi }{4}\right)=0\\ \iff \text{cos }\left(x+\text{\hspace{0.17em}\hspace{0.17em}}\frac{\pi }{4}\right)=0\iff x+\text{\hspace{0.17em}\hspace{0.17em}}\frac{\pi }{4}=\frac{\pi }{2}+\text{}k\pi \iff x=\frac{\pi }{4}+\text{}k\pi \end{array}$

$\frac{\sin 2x}{2}=-2\iff \sin 2x=-4<\text{}-1 nên loại$

Vậy nghiệm của phương trình đã cho là: $x=\frac{\pi }{4}+\text{}k\pi$