Question

Giải pt

\(\sin x+\cos x+\dfrac{1}{\sin x}+\dfrac{1}{\cos x}=\dfrac{10}{3}\)

Answer 1

ĐKXĐ: \(x\ne k\dfrac{\pi}{2}\)

\(sinx+cosx+\dfrac{sinx+cosx}{sinx.cosx}=\dfrac{10}{3}\)

Đặt \(sinx+cosx=t\in\left[-\sqrt{2};\sqrt{2}\right]\)

\(sinx.cosx=\dfrac{t^2-1}{2}\)

Pt trở thành:

\(t+\dfrac{2t}{t^2-1}=\dfrac{10}{3}\)

\(\Rightarrow3t^3-10t^2+3t+10=0\)

\(\Leftrightarrow\left(t-2\right)\left(3t^2-4t-5\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}t=2\left(loại\right)\\t=\dfrac{2-\sqrt{19}}{3}\\t=\dfrac{2+\sqrt{19}}{3}\left(loại\right)\end{matrix}\right.\)

\(\Rightarrow...\)