Question

The complex number $sinx+icos2x\&cosx-isin2x$ are conjugate to each other for :

• A. $x=n\pi$
• B. $x=(n+\frac{1}{2})\pi$
• C. $x=0$
• D. No value of x.

Answer 1

The correct option is D No value of x.

Given $\overline{\sin x+i\cos 2x}=\cos x-i\sin 2x$

$\Rightarrow \sin x-i\cos 2x=\cos x-i\sin 2x$

Comparing the Real and Imaginary parts on either sides

$\sin x=\cos x$ and $\cos 2x=\sin 2x$

$\Rightarrow \tan x=1$ and $\tan 2x=1$

$\Rightarrow x=n\pi +\pi 4$ and $x=n{\pi }^2+\pi 8$

$x$ cannot be simultaneously equal to $2$ different values.

Therefore no value of $x$ exist