\(\left\{{}\begin{matrix}sina-cosa=\sqrt{2}\\sin^2a+cos^2a=1\end{matrix}\right.\)
\(\Rightarrow\left(cosa+\sqrt{2}\right)^2+cos^2a=1\)
\(\Leftrightarrow2cos^2a+2\sqrt{2}cosa+1=0\)
\(\Leftrightarrow\left(\sqrt{2}cosa+1\right)^2=0\)
\(\Leftrightarrow cosa=-\frac{\sqrt{2}}{2}\Rightarrow sina=\frac{\sqrt{2}}{2}\)
\(\Rightarrow sin^5a+cos^5a=0\)