Giải thích các bước giải:
ĐKXĐ: \(\left\{ \begin{array}{l}
\sin x \ne 0\\
\cos x \ne 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x \ne k\pi \\
x \ne \dfrac{\pi }{2} + k\pi
\end{array} \right. \Leftrightarrow x \ne \dfrac{{k\pi }}{2}\)
Ta có:
\(\begin{array}{l}
\tan x = \dfrac{{\sin x}}{{\cos x}} = \dfrac{1}{{\dfrac{{\cos x}}{{\sin x}}}} = \dfrac{1}{{\cot x}}\\
2\tan x - 2\cot x - 3 = 0\\
\Leftrightarrow 2\tan x - \dfrac{2}{{\tan x}} - 3 = 0\\
\Leftrightarrow 2{\tan ^2}x - 2 - 3\tan x = 0\\
\Leftrightarrow 2{\tan ^2}x - 3\tan x - 2 = 0\\
\Leftrightarrow \left( {2\tan x + 1} \right)\left( {\tan x - 2} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\tan x = - \dfrac{1}{2}\\
\tan x = 2
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \arctan \left( { - \dfrac{1}{2}} \right) + k\pi \\
x = \arctan 2 + k\pi
\end{array} \right.\\
x \in \left( {\dfrac{{ - \pi }}{2};2\pi } \right) \Rightarrow có \,\,5\,\,nghiệm\,\,thỏa\,\,\, mãn\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\end{array}\)
