\(sinx-cosx=\frac{1}{5}\Rightarrow\left(sinx-cosx\right)^2=\frac{1}{25}\)
\(\Rightarrow sin^2x+cos^2x-2sinx.cosx=\frac{1}{25}\)
\(\Rightarrow1-sin2x=\frac{1}{25}\)
\(\Rightarrow sin2x=1-\frac{1}{25}=\frac{24}{25}\)
\(sinx-cosx=\frac{1}{5}\)
\(\Leftrightarrow\left(sinx-cosx\right)^2=\frac{1}{25} \)
\(\Leftrightarrow sin^2x+cos^2x-2sinxcosx=\frac{1}{25}\)
\(\Leftrightarrow1-sin2x=\frac{1}{25}\)
\(\Leftrightarrow sin2x=\frac{24}{25}\)