Question

If cosA + sinA = $\sqrt{2}$cosA, then cosA - sinA =

• A. $\sqrt{2}sinA$
• B. $2sinA$
• C. $-\sqrt{2}sinA$
• D. $-2sinA$

Answer 1

The correct option is A $\sqrt{2}sinA$

Given, $\cos A+\sin A=\sqrt{2}\cos A$

$\Rightarrow \sin A=\sqrt{2}\cos A-\cos A$

$\Rightarrow \sin A=(\sqrt{2}-1)\cos A$

$\Rightarrow \frac{\sin A}{\cos A}=\sqrt{2}-1$

$\Rightarrow \tan A=\sqrt{2}-1$

$\Rightarrow \frac{1}{\tan A}=\frac{1}{\sqrt{2}-1}$

$\Rightarrow \cot A=\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$

$\Rightarrow \cot A=\sqrt{2}+1$

$\Rightarrow \frac{\cos A}{\sin A}=\sqrt{2}+1$

$\Rightarrow \cos A=\sin A(\sqrt{2}+1)$

$\Rightarrow \cos A=\sqrt{2}\sin A+\sin A$

$\Rightarrow \cos A-\sin A=\sqrt{2}\sin A$