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Q.
$sin\,x + i \,cos \,2x$ and $cos\, x - i\, sin \,2x$ are conjugate to each other for
Complex Numbers and Quadratic Equations
Solution:
Let $z=sinx+icos2x$
According to the given condition,
$\bar{z}=cosx-isin2x$
$\Rightarrow \, sinx-icos2x=cosx-isin2x$
$\Rightarrow \, \left(sinx-cosx\right)+i\left(sin 2x-cos2x\right)=0$
On equating real and imaginary parts, we get
$sinx - cosx = 0, sin2x - cos2x = 0$
$\Rightarrow \, tanx = 1$ and $tan2x = 1$
$\Rightarrow x=\pi/ 4$ and $x=\pi /8$
which is not possible.