Question

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

• A. $\sqrt{2-k^2}$
• B. $\sqrt{k^2-2}$
• C. $|k|$
• D. $\sqrt{2}-k$
• E. $k-\sqrt{2}$

Answer 1

The correct option is B $\sqrt{k^2-2}$

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

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

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

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

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

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

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

or

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