Question:
If sin x + cos x = a, then sin x – cos x = __________.
Solution:
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 $\sin x-\cos x=\pm \sqrt{2-a^{2}}$
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