$| c o s x | = c o s x - 2 s i n x$ if-

x = n π

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x = 2 n π

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x = n π + π / 4

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x = (2 n + 1) π + π / 4

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Solution

The correct options are
B $x = 2 n π$
D $x = (2 n + 1) π + π / 4$
$| cos x | = cos x - 2 sin x$
For $cos x > 0 \Rightarrow (4 n - 1) \frac{π}{2} < x < (4 n + 1) \frac{π}{2}$
$cos x = cos x - 2 sin x \Rightarrow 2 sin x = 0 \Rightarrow sin x = 0 \Rightarrow x = n π$
$∴ x = 2 n π$
And for $cos x < 0 \Rightarrow (4 n + 1) \frac{π}{2} < x < (4 n + 3) \frac{π}{2}$
$- cos x = cos x - 2 sin x \Rightarrow - 2 cos x + 2 sin x = 0 \Rightarrow cos x + sin x = 0$
$\Rightarrow sin (\frac{π}{4} + x) = 0$
$∴ x = (2 n + 1) π + \frac{π}{4}$

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