The complex numbers sin x + i cos 2x and cos | vedclass

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(D) Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are conjugates if $z_1 = \overline{z_2}$.Given $z_1 = \sin x + i \cos 2x$ and $z_2 = \cos x

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No value of $x$

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(D) Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are conjugates if $z_1 = \overline{z_2}$.Given $z_1 = \sin x + i \cos 2x$ and $z_2 = \cos x - i \sin 2x$.The conjugate of $z_2$ is $\overline{z_2} = \cos x + i \sin 2x$.Equating $z_1 = \overline{z_2}$,we get $\sin x + i \cos 2x = \cos x + i \sin 2x$.Comparing real and imaginary parts:$1$) $\sin x = \cos x$ $\Rightarrow 	an x = 1$ $\Rightarrow x = n\pi + \frac{\pi}{4}$.$2$) $\cos 2x = \sin 2x$ $\Rightarrow 	an 2x = 1$ $\Rightarrow 2x = m\pi + \frac{\pi}{4}$ $\Rightarrow x = \frac{m\pi}{2} + \frac{\pi}{8}$.Since there is no common value of $x$ that satisfies both equations simultaneously,there is no value of $x$ for which these complex numbers are conjugates.

List-$I$	List-$II$
$(i)$ $a \bar{a}$	$(A)$ $-\frac{\pi}{3}$
$(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$	$(B)$ $-i \sqrt{3}$
$(iii)$ $a - \bar{a}$	$(C)$ $2i / \sqrt{3}$
$(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$	$(D)$ $1$
	$(E)$ $\pi / 3$
	$(F)$ $\frac{2}{\sqrt{3}}$

The complex numbers $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other,for

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