If sin x + cos x = a

Given sin x + cos x = a

i.e (sin x – cos x)2 =a2     (squaring both side)

$\Rightarrow \sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=a^{2}$

$\Rightarrow 1+2 \sin x \cos x=a^{2}$

i. e $2 \sin x \cos x=a^{2}-1$

Now, consider

$(\sin x-\cos x)^{2}=\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x$

$=1-\left(a^{2}-1\right)$

$=1-a^{2}+1$

$(\sin x-\cos x)^{2}=2-a^{2}$

i. e $\sin x-\cos x=\pm \sqrt{2-a^{2}}$