Question : If $\cos x+\sin x=\sqrt{2} \cos x$, what is the value of $(\cos x-\sin x)^2+(\cos x+\sin x)^2$?
Option 1: $2$
Option 2: $1$
Option 3: $0$
Option 4: $\frac{1}{\sqrt{2}}$
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Correct Answer: $2$
Solution :
Given:
$\cos x+\sin x=\sqrt{2} \cos x$
To find,
$(\cos x-\sin x)^2+(\cos x+\sin x)^2$
$=\cos^2 x + \sin^2 x - 2\sin x \cos x + \cos^2 x + \sin^2 x + 2\sin x \cos x $
$=1 - 2 \sin x \cos x + 1 + 2\sin x \cos x = 1 + 1 = 2$
Hence, the correct answer is $2$.
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