Question

sin x + i cos 2x and cos x – i sin 2x are conjugate to each other for

(a) x = nπ

(b) $x=\left(n+\frac{1}{2}\right)\frac{\mathrm{\pi }}{2}$

(c) x = 0

(d) No value of x

Answer 1

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.