$\begin{array}{l}\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\pi \theta }\right)=1\\ \Rightarrow \cos \left(\mathrm{\theta }\right)=1\text{ and }\cos \left(\mathrm{\pi \theta }\right)=1\\ \text{ or }\cos \left(\mathrm{\theta }\right)=-1\text{ and }\cos \left(\mathrm{\pi \theta }\right)=-1\\ \text{ If }\cos \left(\mathrm{\theta }\right)=1\Rightarrow \mathrm{\theta }=2\mathrm{m\pi }\text{ and }\cos \left(\mathrm{\pi \theta }\right)=1\Rightarrow \mathrm{\theta }=2\mathrm{\pi }\text{ which is possible only when }\mathrm{\theta }=0.\\ \text{ Equation (ii) is not possible for any }\mathrm{\theta }\text{ as for }\cos \left(\mathrm{\theta }\right)=-1,\mathrm{\theta }\text{ should be odd multiple of }\mathrm{\pi }\Rightarrow \text{ irrational }\\ \text{ and for }\cos \left(\mathrm{\pi \theta }\right)=-\mathrm{l}\Rightarrow \mathrm{\theta }\text{ should be odd integer }\Rightarrow \text{ rational }\\ \text{ Both the conditions cannot be satisfied. }\\ \text{ Therefore, }\mathrm{\theta }=0\text{ is the only solution. }\end{array}$