Cho biết \(\sin \alpha -\cos \alpha =\frac{1}{\sqrt{5}}.\) Giá trị của \(P=\sqrt{{{\sin }^{4}}\alpha +{{\cos }^{4}}\alpha }\) bằng?

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

\(\Leftrightarrow 1-2\sin \alpha \cos \alpha =\frac{1}{5}\Leftrightarrow \sin \alpha \cos \alpha =\frac{2}{5}\)

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

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