$$\frac{1+i \cos \theta}{1-2 i \cos \theta}$$ 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 $$\theta=3 \pi$$