Đáp án:
\(\left[ \begin{array}{l}
x = \dfrac{\pi }{4} + k\pi \\
x = - \dfrac{\pi }{4} + k\pi \\
x = - \dfrac{\pi }{3} + k\pi
\end{array} \right.\)
Giải thích các bước giải:
\(\begin{array}{l}
{\sin ^3}x - \sqrt 3 .{\cos ^3}x = \sin x.{\cos ^2}x - \sqrt 3 .{\sin ^2}x.\cos x\\
\to \dfrac{{{{\sin }^3}x}}{{{{\cos }^3}x}} - \dfrac{{\sqrt 3 .{{\cos }^3}x}}{{{{\cos }^3}x}} = \dfrac{{\sin x.{{\cos }^2}x}}{{{{\cos }^3}x}} - \dfrac{{\sqrt 3 .{{\sin }^2}x.\cos x}}{{{{\cos }^3}x}}\\
\to {\tan ^3}x - \sqrt 3 = \tan x - \sqrt 3 {\tan ^2}x\\
\to {\tan ^3}x + \sqrt 3 {\tan ^2}x - \tan x - \sqrt 3 = 0\\
\to \left[ \begin{array}{l}
\tan x = 1\\
\tan x = - 1\\
\tan x = - \sqrt 3
\end{array} \right.\\
\to \left[ \begin{array}{l}
x = \dfrac{\pi }{4} + k\pi \\
x = - \dfrac{\pi }{4} + k\pi \\
x = - \dfrac{\pi }{3} + k\pi
\end{array} \right.\left( {k \in Z} \right)
\end{array}\)
