$(sin\alpha+cos\alpha)^2= m^2$
$\Leftrightarrow sin^2\alpha + cos^2\alpha + 2sin\alpha.cos\alpha= m^2$
$\Leftrightarrow 2sin\alpha.cos\alpha = m^2-1$
$(sin\alpha - cos\alpha)^2= sin^2\alpha - 2sin\alpha.cos\alpha + cos^2\alpha$
$= 1-(m^2-1)= 2-m^2$
$\Leftrightarrow sin\alpha - cos\alpha= \sqrt{2-m^2}$