1) $2\sin 4x-1=0$
$\Rightarrow \sin 4x=\dfrac{1}{2}$
$\Rightarrow \left[ \begin{array}{l} 4x=\dfrac{\pi}{6}+k2\pi \\ 4x=\pi-\dfrac{\pi}{6}+k2\pi\end{array} \right .$
$\Rightarrow \left[ \begin{array}{l} x=\dfrac{\pi}{24}+k\dfrac{\pi}{2} \\x=\dfrac{5\pi}{24}+k\dfrac{\pi}{2}\end{array} \right .(k\in\mathbb Z)$
2) $y=\dfrac{\cot x+1}{1+{\tan}^2x}$
Đk: $\left\{ \begin{array}{l} \sin x\ne0 \\ \cos x\ne0 \end{array} \right .$
$\Leftrightarrow\sin 2x\ne0$
$\Leftrightarrow 2x\ne k\pi$
$\Leftrightarrow x\ne k\dfrac{\pi}{2}(k\in\mathbb Z)$
3) $y=\cos\sqrt{x}+\dfrac{1}{x}$
Đk: $\left\{ \begin{array}{l} x\ge0\\ x\ne0 \end{array} \right .$
$\Leftrightarrow x>0$
