HMMT 2015 U20

What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that

$$\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$$

and

$$\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ?$$