Cho biết

C

- Hướng dẫn giải

Ta có $2\cos \alpha +\sqrt{2}\sin \alpha =2\iff \sqrt{2}\sin \alpha =2-2\cos \alpha \to 2{\sin }^2\alpha ={\left(2-2\cos \alpha \right)}^2$

$\begin{array}{l}\iff 2{\sin }^2\alpha =4-8\cos \alpha +4{\cos }^2\alpha \iff 2\left(1-{\cos }^2\alpha \right)=4-8\cos \alpha +4{\cos }^2\alpha \\ \iff 6{\cos }^2\alpha -8\cos \alpha +2=0\iff \left[\begin{array}{l}\cos \alpha =1\\ \cos \alpha =\frac{1}{3}\end{array}\right..\end{array}$

 $\cos \alpha =1$: không thỏa mãn vì $0^0<\alpha <{90}^0.$

$\cos \alpha =\frac{1}{3}\Rightarrow \sin \alpha =\frac{2\sqrt{2}}{3}\Rightarrow \cot \alpha =\frac{\cos \alpha }{\sin \alpha }=\frac{\sqrt{2}}{4}.$ 

Chọn C.