\(\dfrac{2sin^2x-sinx}{1-cos2x}=0\) (x≠\(\dfrac{k\pi}{2}\))
\(\Leftrightarrow\dfrac{2sin^2x-sinx}{1-1+sin^2x}=0\)
\(\Leftrightarrow\dfrac{sin^2x\left(2-\dfrac{1}{sinx}\right)}{sin^2x}=0\)
\(\Leftrightarrow2-\dfrac{1}{sinx}=0\)
\(\Rightarrow sinx=\dfrac{1}{2}=sin\left(\dfrac{\pi}{6}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
*Trường hợp 1: \(x=\dfrac{\pi}{6}+k2\pi\)
\(-2010\pi< \dfrac{\pi}{6}+k2\pi< 2020\pi\)
=> \(-\dfrac{12061}{12}< k< \dfrac{12119}{12}\)
=>k∈{-1005;-1004;...;1009}
=> có 2015 nghiệm nguyên(1)
*trường hợp 2: \(x=\dfrac{5\pi}{6}+k2\pi\)
\(-2010\pi< \dfrac{5\pi}{6}+k2\pi< 2020\pi\)
=>\(-\dfrac{12065}{12}< k< \dfrac{12115}{12}\)
=>k∈{-1005;-1004;...;1009}
=> có 2015 nghiệm nguyên (2)
Từ (1) và (2) suy ra: trong khoảng \(\left(-2010\pi;2020\pi\right)\) có 4030 nghiệm nguyên
