Cho biết

B

- Hướng dẫn giải

Ta có $\sin \alpha -\cos \alpha =\frac{1}{\sqrt{5}}\Rightarrow {\left(\sin \alpha -\cos \alpha \right)}^2=\frac{1}{5}$

$\iff 1-2\sin \alpha \cos \alpha =\frac{1}{5}\iff \sin \alpha \cos \alpha =\frac{2}{5}.$

Ta có $P=\sqrt{{\sin }^4\alpha +{\cos }^4\alpha }=\sqrt{{\left({\sin }^2\alpha +{\cos }^2\alpha \right)}^2-2{\sin }^2\alpha {\cos }^2\alpha }$

$=\sqrt{1-2{\left(\sin \alpha cos\alpha \right)}^2}=\frac{\sqrt{17}}{5}.$ 

Chọn B.