ĐKXĐ: \(\left\{{}\begin{matrix}sinx< >0\\sin2x< >0\\sin4x< >0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x< >k\Omega\\2x< >k\Omega\\4x< >k\Omega\end{matrix}\right.\Leftrightarrow x\ne\dfrac{k\Omega}{4}\)
\(\dfrac{1}{sinx}+\dfrac{1}{sin2x}+\dfrac{1}{sin4x}=0\)
=>\(\dfrac{1}{sinx}+cotx+\dfrac{1}{sin2x}+cot2x+\dfrac{1}{sin4x}+cot4x=cotx+cot2x+cot4x\)
=>\(\dfrac{1+cosx}{sinx}+\dfrac{1+cos2x}{sin2x}+\dfrac{1+cos4x}{sin4x}=cotx+cot2x+cot4x\)
=>\(\dfrac{2\cdot cos^2\left(\dfrac{x}{2}\right)}{2\cdot sin\left(\dfrac{x}{2}\right)\cdot cos\left(\dfrac{x}{2}\right)}+\dfrac{2\cdot cos^2x}{2\cdot sinx\cdot cosx}+\dfrac{2\cdot cos^22x}{2\cdot sin2x\cdot cos2x}=cotx+cot2x+cot4x\)
=>\(\dfrac{cos\left(\dfrac{x}{2}\right)}{sin\left(\dfrac{x}{2}\right)}+\dfrac{cosx}{sinx}+\dfrac{cos2x}{sin2x}=cotx+cot2x+cot4x\)
=>\(cot\left(\dfrac{x}{2}\right)+cotx+cot2x=cotx+cot2x+cot4x\)
=>\(cot4x=cot\left(\dfrac{x}{2}\right)\)
=>\(\left\{{}\begin{matrix}4x=\dfrac{x}{2}+k\Omega\\4x< >k\Omega\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\dfrac{7}{2}x=k\Omega\\x< >\dfrac{k\Omega}{4}\end{matrix}\right.\Leftrightarrow x=\dfrac{2}{7}k\Omega\)
