Question

The complex numbers $sinx+icos2x$ and $cosx-isin2x$ are conjugate to each other for

• A.
• B.
• C. x=0
• D. No value of x

Answer 1

The correct option is D No value of x

$sinx+icos2x$ and $cosx-isin2x$ are conjugate to each other if $sinx=cosx$ and $cos2x=sin2x$
or $tanx=1\Rightarrow x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},......$ ......(i)
and $tan2x=1\Rightarrow 2x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4}......$
or $x=\frac{\pi }{8},\frac{5\pi }{8},\frac{9\pi }{8}..........\left(ii\right)$

There exists no value of $x$ common in (i) and (ii). Therefore there is no value of $x$ for which
the given complex numbers are conjugate.