Cho biết

B

- Hướng dẫn giải

Ta có $\cos \alpha +\sin \alpha =\frac{1}{3}\Rightarrow {\left(\cos \alpha +\sin \alpha \right)}^2=\frac{1}{9}$

$\iff 1+2\sin \alpha \cos \alpha =\frac{1}{9}\iff \sin \alpha \cos \alpha =-\frac{4}{9}.$

Ta có 

$P=\sqrt{{\tan }^2\alpha +{\cot }^2\alpha }=\sqrt{{\left(\tan \alpha +\cot \alpha \right)}^2-2\tan \alpha \cot \alpha }=\sqrt{{\left(\frac{\sin \alpha }{\cos \alpha }+\frac{\cos \alpha }{\sin \alpha }\right)}^2-2}$

$=\sqrt{{\left(\frac{{\sin }^2\alpha +{\cos }^2\alpha }{\sin \alpha \cos \alpha }\right)}^2-2}=\sqrt{{\left(\frac{1}{\sin \alpha \cos \alpha }\right)}^2-2}=\sqrt{{\left(-\frac{9}{4}\right)}^2-2}=\frac{7}{4}.$ 

Chọn B.