ĐKXĐ: \(x\ne\dfrac{k\pi}{2}\)
\(\dfrac{sinx}{cosx}+\dfrac{cosx}{sinx}+7=\dfrac{cos^22x}{sin^22x}\)
\(\Leftrightarrow\dfrac{sin^2x+cos^2x}{sinx.cosx}+7=\dfrac{1-sin^22x}{sin^22x}\)
\(\Leftrightarrow\dfrac{2}{sin2x}+7=\dfrac{1}{sin^22x}-1\)
\(\Leftrightarrow\dfrac{1}{sin^22x}-\dfrac{2}{sin2x}-8=0\)
\(\Rightarrow\left[{}\begin{matrix}\dfrac{1}{sin2x}=4\\\dfrac{1}{sin2x}=-2\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}sin2x=\dfrac{1}{4}\\sin2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}arcsin\left(\dfrac{1}{4}\right)+k\pi\\x=\dfrac{\pi}{2}-\dfrac{1}{2}arcsin\left(\dfrac{1}{4}\right)+k\pi\\x=-\dfrac{\pi}{12}+k\pi\\x=\dfrac{7\pi}{12}+k\pi\end{matrix}\right.\)
