1) $3\cos x -\sin x=3$
$\Rightarrow \dfrac{3}{\sqrt{10}}\cos x-\dfrac{1}{\sqrt{10}}\sin x=\dfrac{3}{\sqrt{10}}$
Đặt $\sin\alpha=\dfrac{3}{\sqrt{10}}$ và $\cos\alpha=\dfrac{1}{\sqrt{10}}$
Phương trình tương đương: $\sin\alpha\cos x-\cos\alpha\sin x=\dfrac{3}{\sqrt{10}}$
$\Rightarrow \sin(\alpha-x)=\dfrac{3}{\sqrt{10}}$
$\Rightarrow \left[\begin{array}{l} \alpha-x=\arcsin\dfrac{3}{\sqrt{10}}+k2\pi\\ \alpha-x=\pi-\arcsin\dfrac{3}{\sqrt{10}}+k2\pi \end{array} \right .$
$\Rightarrow \left[\begin{array}{l} x=\alpha-\arcsin\dfrac{3}{\sqrt{10}}+k2\pi\\ x=\alpha-\pi+\arcsin\dfrac{3}{\sqrt{10}}+k2\pi \end{array} \right .$
2) $5{\cos}^2x-3\cos x-2=0$
$\Rightarrow \left[ \begin{array}{l} \cos x=-1 \\ \cos x=\dfrac{2}{5} \end{array} \right .$
$\Rightarrow \left[ \begin{array}{l} x=\pi+k2\pi \\x=\pm\arccos\dfrac{2}{5}+k2\pi \end{array} \right .(k\in\mathbb Z)$
3) ${\cos}^2x-{\sin}^2x\cos x=0$
$\Rightarrow \cos x(\cos x-{\sin}^2x)$
$\Rightarrow \left\{ \begin{array}{l} \cos x=0\Rightarrow x=\dfrac{\pi}{2}+k\pi(k\in\mathbb Z)\\ \cos x-{\sin}^2x=0(1)\end{array} \right .$
$(1)\Rightarrow \cos x-(1-{\cos}^2x)=0$
$\Rightarrow \left[\begin{array}{l} \cos x=\dfrac{-1-\sqrt5}{2} <-1(l)\\ \cos x=\dfrac{-1-\sqrt5}{2} \Rightarrow x=\pm\arccos\dfrac{-1-\sqrt5}{2}+k2\pi(k\in\mathbb Z)\end{array} \right .$
