If cos2A = sin2A then sinA + cosA is

$\begin{array}{l}\cos 2\mathrm{A}=\sin 2\mathrm{A}\\ {\cos }^2\mathrm{A}-{\sin }^2\mathrm{A}=2\sin \mathrm{Acos}\mathrm{A}\\ {\sin }^2\mathrm{A}+2\sin \mathrm{Acos}\mathrm{A}={\cos }^2\mathrm{A}\end{array}$
Adding cos²A on both sides
$\begin{array}{c}{\sin }^2\mathrm{A}+\mathrm{sinAcos}\mathrm{A}+\cos 2=2{\cos }^2\mathrm{A}\\ (\sin \mathrm{A}+\cos \mathrm{A}{)}^2=2{\cos }^2\mathrm{A}\\ \sin \mathrm{\theta }+\cos \mathrm{\theta }=\pm \sqrt{2}\cos \mathrm{A}\end{array}$