Question

If sin x + cos x = a, then sin x – cos x = __________.

Answer 1

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

$$\begin{gathered} \Rightarrow {\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x=a^2 \\ \Rightarrow 1+2\sin x\cos x=a^2 \\ \mathrm{i}.\mathrm{e}2\sin x\cos x=a^2-1 \end{gathered}$$
Now, consider
$\begin{array}{rcl}{\left(\sin x-\cos x\right)}^2& =& {\sin }^{2}x+{\cos }^{2}x-2\sin x\cos x\\ & =& 1-\left(a^2-1\right)\\ & =& 1-a^2+1\\ {\left(\sin x-\cos x\right)}^2& =& 2-a^2\\ \mathrm{i}.\mathrm{e}\sin x-\cos x& =& \pm \sqrt{2-a^2}\end{array}$