Đáp án:
Có tất cả 4034 nghiệm.
Giải thích các bước giải:
\(\begin{array}{l} 4{\tan ^2}x - 5\tan x + 1 = 0\,\,\,\,\left( * \right)\\ ĐK:\,\,\,\cos x \ne 0\\ \left( * \right) \Leftrightarrow \left[ \begin{array}{l} \tan x = 1\\ \tan x = \dfrac{1}{4} \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{\pi }{4} + k\pi \\ x = \arctan \dfrac{1}{4} + m\pi \end{array} \right.\,\,\,\left( {k \in Z} \right)\\ x \in \left( { - \dfrac{{2017\pi }}{2};\,\,\dfrac{{2017\pi }}{2}} \right)\\ \Rightarrow \left[ \begin{array}{l} - \dfrac{{2017\pi }}{2} < \,\dfrac{\pi }{4} + k\pi < \dfrac{{2017\pi }}{2}\\ - \dfrac{{2017\pi }}{2} < \,\arctan \dfrac{1}{4} + m\pi < \dfrac{{2017\pi }}{2} \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} - 1008,75 < k < 1008,25\\ - 1013,46 < m < 1003,53 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} k \in \left\{ { - 1008; - 1007;.....;\,\,1007;\,\,1008} \right\}\\ m \in \left\{ { - 1013;\,\, - 1012;\,\,....;\,\,1002;\,\,1003} \right\} \end{array} \right.\\ \Rightarrow Có\,\,\,1008.2 + 1 + \left( {1003 + 1013} \right) + 1 = 4034\,\,\,nghiệm. \end{array}\)
