A) \[x=n\pi \]
B) \[x=\left( n+\frac{1}{2} \right)\pi \]
C) \[x=0\]
D) No value of \[x\]
Correct Answer: D
Solution :
Given, numbers are conjugate to each other, \[\therefore \] \[\sin x+i\cos 2x=\cos x-i\sin 2x\] Equating real and imaginary parts, we get \[\sin x=\cos x\]and\[\cos 2x=\sin 2x\] \[\therefore \] \[\tan x=1\] \[\Rightarrow \] \[x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},\] ?(i) and \[\tan 2x=1\] \[\Rightarrow \]\[2x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},\] ?(ii) \[\Rightarrow \]\[x=\frac{\pi }{8},\frac{5\pi }{8},\frac{9\pi }{8},...\] There exists no value of \[x\]common in Eqs. (i) and (ii).
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