Question

If $\cos \theta -\sin \theta =\sqrt{2}\sin \theta$, then $\cos \theta +\sin \theta$ is equal to ?

• A. $\sqrt{2}\cos \theta$
• B. $\sqrt{2}\sin \theta$
• C. $2\cos \theta$
• D. $-\sqrt{2}\cos \theta$

Answer 1

The correct option is A $\sqrt{2}\cos \theta$

$\cos \theta -\sin \theta =\sqrt{2}\sin \theta$

Squaring on both sides.

${\cos }^2\theta +{\sin }^2\theta -2\cos \theta \sin \theta =2{\sin }^2\theta$

Adding ${\cos }^2\theta$ on both sides.

${\cos }^2\theta +{\cos }^2\theta ={\sin }^2\theta +2\cos \theta \sin \theta +{\cos }^2\theta$

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

$\Rightarrow \underset{–}{\sin \theta +\cos \theta =\sqrt{2}\cos \theta }$