The sum of all possible values of $ heta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos heta}{1-2 i \cos heta}$ is purely imaginary, is equal

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The sum of all possible values of $	heta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos 	heta}{1-2 i \cos 	heta}$ is purely imaginary, is equal

Aicharya · Accepted Answer

$\begin{aligned} & Z=\frac{1+i \cos 	heta}{1-2 i \cos 	heta} \ & Z=-\bar{Z} \Rightarrow \frac{1+i \cos 	heta}{1-2 i \cos 	heta}=-\left(\frac{\overline{1+i \cos 	heta}}{1-2 i \cos 	heta}ight) \ & (1+i \cos 	heta)(\overline{1-2 i \cos 	heta})=-(1-2 i \cos 	heta)(\overline{1+i \cos 	heta}) \ & (1+i \cos 	heta)(1+2 i \cos 	heta)=-(1-2 i \cos 	heta)(1-i \cos 	heta) \ & 1+3 i \cos 	heta-2 \cos ^2 	heta=-\left(1-3 i \cos 	heta-2 \cos ^2 	hetaight) \ & 2-4 \cos ^2 	heta=0 \ & \Rightarrow \cos ^2 	heta=\frac{1}{2} \Rightarrow 	heta=-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4} \ & 	ext { sum }=3 \pi\end{aligned}$

The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos \theta}{1-2 i \cos…

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