Question

If $\sin x+\cos x=a$, find the value of $\left|\sin x-\cos x\right|$.

Answer 1

Given: $\sin x+\cos x=a$

Now,

$$\begin{gathered} {\left(\sin x+\cos x\right)}^2+{\left(\sin x-\cos x\right)}^2={\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x+{\sin }^{2}x+{\cos }^{2}x-2\sin x\cos x \\ \Rightarrow {\left(\sin x+\cos x\right)}^2+{\left(\sin x-\cos x\right)}^2=2\left({\sin }^{2}x+{\cos }^{2}x\right) \\ \Rightarrow {\left(\sin x+\cos x\right)}^2+{\left(\sin x-\cos x\right)}^2=2 \end{gathered}$$

$$\begin{gathered} \therefore a^2+{\left(\sin x-\cos x\right)}^2=2 \\ \Rightarrow {\left(\sin x-\cos x\right)}^2=2-a^2 \\ \Rightarrow \sqrt{{\left(\sin x-\cos x\right)}^2}=\sqrt{2-a^2} \\ \Rightarrow \left|\sin x-\cos x\right|=\sqrt{2-a^2}\left(\sqrt{x^2}=\left|x\right|\right) \end{gathered}$$

Thus, the required value is $\sqrt{2-a^2}$.