Explanation

(b)
Since, $\cos{2x,\frac{1}{2},\sin{2x}}$ are in AP
$\Rightarrow \ \cos{2x + \sin{2x = 1}}$
$\Rightarrow \ \sin{2x = 1 - \cos{2x = 2\sin^{2}x}}$
$\Rightarrow \ 2\sin{x(\cos{x - \sin{x) = 0}}}$
$\Rightarrow \ \sin{x = 0\ or\ \cos{x - \sin{x = 0}}}$
$\Rightarrow \ x = n\pi\ or\ \tan{x = 1}$
$\Rightarrow \ x = n\pi\ or\ x = n\pi + \frac{\pi}{4}$
Thus, required values of $x$ are $n\pi$ and $n\pi + \frac{\pi}{4}$