Cho \(\sin x + \cos x = \frac{1}{5}\). Tính \(P = \left| {\sin x - \cos x} \right|\).

* Hướng dẫn giải

\(\begin{array}{l}
\sin x + \cos x = \frac{1}{5} \Rightarrow {\left( {\sin x + \cos x} \right)^2} = \frac{1}{{25}}\\
 \Leftrightarrow {\sin ^2}x + {\cos ^2}x + 2\sin x\cos x = \frac{1}{{25}}\\
 \Leftrightarrow 1 + 2\sin x\cos x = \frac{1}{{25}}\\
 \Leftrightarrow 2\sin x\cos x = \frac{1}{{25}} - 1 = \frac{{ - 24}}{{25}}\\
 \Rightarrow {\left( {\sin x - \cos x} \right)^2} = {\sin ^2}x + {\cos ^2}x - 2\sin x\cos x = 1 - \frac{{ - 24}}{{25}} = \frac{{49}}{{25}}\\
 \Rightarrow P = \left| {\sin x - \cos x} \right| = \sqrt {{{\left( {\sin x - \cos x} \right)}^2}}  = \frac{7}{5}
\end{array}\)