The sum of all possible values of \[\theta \in[-\pi, 2 \pi]\], for which \[\frac{1+i \cos \theta}{1-2 i \cos \theta}\] is purely imaginary, is equal to :
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Official solution
\[\frac{1+i \cos \theta}{1-2 i \cos \theta}\] is purely imaginary
\[n=\frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}=\frac{1+3 i \cos \theta-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}\]
\[n=\frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}+i\left(\frac{3 \cos \theta}{1+4 \cos ^2 \theta}\right)\]
n is purely imaginary
\[\Rightarrow \frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}=0\]
\[\Rightarrow \cos ^2 \theta=\frac{1}{2}\]
\[\Rightarrow \cos \theta= \pm \frac{1}{\sqrt{2}}\]
\[\theta\] can be \[\frac{\pi}{4}, \frac{-\pi}{4}, \frac{3 \pi}{4}, \frac{-3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\]
Sum of all possible values of \[\theta=3 \pi\]