ĐK: \(x\ne k\pi\)
\(3tan^2\left(x-\dfrac{\pi}{2}\right)=2.\dfrac{1-sinx}{sinx}\)
\(\Leftrightarrow3cot^2x=\dfrac{2}{sinx}-2\)
\(\Leftrightarrow\dfrac{3}{sin^2x}-3=\dfrac{2}{sinx}-2\)
\(\Leftrightarrow\dfrac{3}{sin^2x}-\dfrac{2}{sinx}-1=0\)
\(\Leftrightarrow\left(\dfrac{1}{sinx}-1\right)\left(\dfrac{3}{sinx}+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{1}{sinx}-1=0\\\dfrac{3}{sinx}+1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=-3\left(l\right)\end{matrix}\right.\)
\(sinx=1\Leftrightarrow x=\dfrac{\pi}{2}+\text{k}2\pi\left(tm\right)\)
Vậy phương trihf đã cho có nghiệm \(x=\dfrac{\pi}{2}+\text{k}2\pi\)
ĐKXĐ: \(x\ne k\pi\)
\(3cot^2x=2\left(\dfrac{1-sinx}{sinx}\right)\)
\(\Leftrightarrow\dfrac{3cos^2x}{sin^2x}=2\left(\dfrac{1-sinx}{sinx}\right)\)
\(\Leftrightarrow\dfrac{3\left(1-sinx\right)\left(1+sinx\right)}{sin^2x}-2\left(\dfrac{1-sinx}{sinx}\right)=0\)
\(\Leftrightarrow\left(\dfrac{1-sinx}{sinx}\right)\left(\dfrac{3+3sinx}{sinx}-2\right)=0\)
\(\Leftrightarrow\left(\dfrac{1-sinx}{sinx}\right)\left(\dfrac{3+sinx}{sinx}\right)=0\)
\(\Leftrightarrow sinx=1\)
\(\Rightarrow x=\dfrac{\pi}{2}+k2\pi\)
