Đk:\(x\ne\dfrac{k\pi}{2};k\in Z\)
\(sinx+cosx+\dfrac{1}{cosx}+\dfrac{1}{sinx}=\dfrac{10}{3}\)
\(\Leftrightarrow sinx+cosx+\dfrac{sinx+cosx}{sinx.cosx}=\dfrac{10}{3}\)
Đặt \(t=sinx+cosx;t\in\left[-\sqrt{2};\sqrt{2}\right]\)
\(\Rightarrow\dfrac{t^2-1}{2}=sinx.cosx\)
Pttt:\(t+\dfrac{t}{\dfrac{t^2-1}{2}}=\dfrac{10}{3}\)
\(\Leftrightarrow t+\dfrac{2t}{t^2-1}=\dfrac{10}{3}\)\(\Leftrightarrow t^3-\dfrac{10}{3}t^2+t+\dfrac{10}{3}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}t=2\left(ktm\right)\\t=\dfrac{2+\sqrt{19}}{3}\left(ktm\right)\\t=\dfrac{2-\sqrt{19}}{3}\left(tm\right)\end{matrix}\right.\)\(\Rightarrow sinx+cosx=\dfrac{2-\sqrt{19}}{3}\)
\(\Leftrightarrow cos\left(x-\dfrac{\pi}{4}\right)=\dfrac{-\sqrt{38}+2\sqrt{2}}{6}\)
\(\Leftrightarrow x=\dfrac{\pi}{4}\pm arc.cos\left(\dfrac{-\sqrt{38}+2\sqrt{2}}{6}\right)+k2\pi\) \(\left(k\in Z\right)\)(tm)
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