\(3tan3x+cot2x=2tanx+\dfrac{2}{sin4x}\)
Đk: \(\left\{{}\begin{matrix}cos3x\ne0\\sin2x\ne0\\sin4x\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x\ne\dfrac{\pi}{6}+m\pi\\x\ne n\dfrac{\pi}{2}\\x\ne p\dfrac{\pi}{4}\end{matrix}\right.\)
Pt:\(\Rightarrow\)\(3tan3x=2tanx+\dfrac{2}{sin4x}-cot2x\)
\(\Rightarrow3tan3x=2tanx+\dfrac{2}{2sin2x\cdot cos2x}-\dfrac{cos2x}{sin2x}\)
\(\Rightarrow3tan3x=2tanx+\dfrac{1-cos^22x}{sin2x\cdot cos2x}\)
\(\Rightarrow\)\(3tan3x=2tanx+\dfrac{sin^22x}{sin2x\cdot cos2x}=2tanx+tan2x\)
\(\Rightarrow\)\(3tan3x-3tanx=-tanx+tan2x\)
\(\Rightarrow\)\(3\left(\dfrac{sin3x}{cos3x}-\dfrac{sinx}{cosx}\right)-\left(\dfrac{sin2x}{cos2x}-\dfrac{sinx}{cosx}\right)=0\)
\(\Rightarrow\)\(3\dfrac{sin3x\cdot cosx-cos3x\cdot sinx}{cos3x\cdot cosx}-\dfrac{sin2x\cdot cosx-sinx\cdot cos2x}{cos2x\cdot cox}=0\)
\(\Rightarrow3\dfrac{sin2x}{cos3x\cdot cosx}-\dfrac{sinx}{cos2x\cdot cosx}=0\)
\(\Rightarrow3sin2x\cdot cos2x-sinx\cdot cos3x=0\)
\(\Rightarrow3sin4x-sin2x-sin4x=0\)
\(\Rightarrow\)2sin4x-sin2x=0\(\Rightarrow4sin2x\cdot cos2x-sin2x=0\)
\(\Rightarrow\left[{}\begin{matrix}sin2x=0\\4cos2x-1=0\end{matrix}\right.\) (loại sin2x=0)
Suy ra 4cos2x-1=0
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