Find the real values of θ for which the complex number

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

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

$=\frac{1+2 i \cos \theta+i \cos \theta-2 \cos \theta}{1+4 \cos ^{2} \theta}$

$=\frac{1-2 \cos \theta+i 3 \cos \theta}{1+4 \cos ^{2} \theta}$

For it to be purely real, the imaginary part must be zero. $3 \cos \theta=0$

This is true for odd multiples of $\frac{\pi}{2}$.

$\therefore \theta=(2 n+1) \frac{\pi}{2}, n \in Z$