11.
\(\left(2cosx+5\right)\left(sin^4\dfrac{x}{2}-cos^4\dfrac{x}{2}\right)+3=0\)
\(\Leftrightarrow\left(2cosx+5\right)\left(sin^2\dfrac{x}{2}-cos^2\dfrac{x}{2}\right)+3=0\)
\(\Leftrightarrow\left(2cosx+5\right)\left(-cosx\right)+3=0\)
\(\Leftrightarrow-2cos^2x-5cosx+3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{1}{2}\\cosx=-3\left(vn\right)\end{matrix}\right.\)\(\Rightarrow x=\pm\dfrac{\pi}{3}+k2\pi\) \(\left(k\in Z\right)\)
\(x\in\left(0;2\pi\right)\Rightarrow\)\(\left[{}\begin{matrix}0< -\dfrac{\pi}{3}+k2\pi< 2\pi\\0< \dfrac{\pi}{3}+k2\pi< 2\pi\end{matrix}\right.\)\(\left(k\in Z\right)\)\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{1}{6}< k< \dfrac{7}{6}\\-\dfrac{1}{6}< k< \dfrac{5}{6}\end{matrix}\right.\)\(\left(k\in Z\right)\)\(\Rightarrow\left[{}\begin{matrix}k=1\\k=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{2\pi}{3}\\x=\dfrac{\pi}{3}\end{matrix}\right.\)
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