If $\sin \alpha + \cos \alpha = k$, then $|\sin \alpha - \cos \alpha |$ equals.

$\sin\alpha +\cos\alpha =k$

${ \left( \sin\alpha +\cos\alpha \right) }^{ 2 }={ k }^{ 2 }$

${ \sin }^{ 2 }\alpha +{ \cos }^{ 2 }\alpha +2\sin\alpha \cos\alpha ={ k }^{ 2 }$

$2\sin\alpha \cos\alpha ={ k }^{ 2 }-1$

${ \left| \sin\alpha -\cos\alpha \right| }^{ 2 }=\left| { \sin }^{ 2 }\alpha +{ \cos }^{ 2 }\alpha -2\sin\alpha \cos\alpha \right|$

${ \left| \sin\alpha -\cos\alpha \right| }^{ 2 }=\left| 1-\left( { k }^{ 2 }-1 \right) \right|$

${ \left| \sin\alpha -\cos\alpha \right| }^{ 2 }=\left| 2-{ k }^{ 2 } \right|$

or

$\left| \sin\alpha -\cos\alpha \right| =\sqrt { 2-{ k }^{ 2 } }$