Question:
sin x + i cos 2x and cos x – i sin 2x are conjugate to each other for
(a) x = nπ
(b) $x=\left(n+\frac{1}{2}\right) \frac{\pi}{2}$
(c) x = 0
(d) No value of x
Solution:
Given sin x + i cos 2x and cos x – i sin 2x are conjugate to each other
i.e sin x + i cos 2x = cos x – i sin 2x
i.e sin x – i cos 2x = cos x – i sin 2x
on comparing real and imaginary part,
sin x = cos x and cos 2x = sin 2x
i.e. sin x = cos x and 2cos2 x – 1 = 2 sin x cos x
i.e 2cos2 x – 1 = 2 cos x cos x (∴ sin x = cos x)
i.e 2cos2 x – 1 = 2cos2 x
i.e – 1 = 0
which is a false statement.
Hence no value of x exist
Therefore, the correct answer is option D.
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