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

$ Z=-\bar{Z} \Rightarrow \frac{1+i \cos \theta}{1-2 i \cos \theta}=-\left(\frac{\overline{1+i \cos \theta}}{1-2 i \cos \theta}\right) $

$ (1+i \cos \theta)(\overline{1-2 i \cos \theta})=-(1-2 i \cos \theta)(\overline{1+i \cos \theta}) $

$ (1+i \cos \theta)(1+2 i \cos \theta)=-(1-2 i \cos \theta)(1-i \cos \theta) $

$ 1+3 i \cos \theta-2 \cos ^2 \theta=-\left(1-3 i \cos \theta-2 \cos ^2 \theta\right) $

$ 2-4 \cos ^2 \theta=0 $

$ \Rightarrow \cos ^2 \theta=\frac{1}{2} \Rightarrow \theta=-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4} $

$ \text { sum }=3 \pi$