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 :

<p>$\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary</p> <p>$$n=\frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}=\frac{1+3 i \cos \theta-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}$$</p> <p>$$n=\frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}+i\left(\frac{3 \cos \theta}{1+4 \cos ^2 \theta}\right)$$</p> <p>$n$ is purely imaginary</p> <p>$\Rightarrow \frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}=0$</p> <p>$\Rightarrow \cos ^2 \theta=\frac{1}{2}$</p> <p>$\Rightarrow \cos \theta= \pm \frac{1}{\sqrt{2}}$</p> <p>$\theta$ can be $$\frac{\pi}{4}, \frac{-\pi}{4}, \frac{3 \pi}{4}, \frac{-3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$$</p> <p>Sum of all possible values of $\theta=3 \pi$</p>