I am afraid I am missing something, but let me try nonetheless. Take $\phi=\theta+r\pi$ for any rational $r\in(0,\frac{1}{2}).$ We are trying to find a solution of the equation $$f(\theta)=\cos^2(\theta)+cos(\theta+r\pi)=0.$$ Clearly, $f(0)>0,$ and $f(\frac{\pi}{2})<0,$ so it has a solution.

I am afraid I am missing something, but let me try nonetheless. Take $\phi=\theta+r\pi$ for any rational $r\in(0,\frac{1}{2}).$ We are trying to find a solution of the equation $$f(\theta)=\cos^2(\theta)+\cos(\theta+r\pi)=0.$$ Clearly, $f(0)>0,$ and $f(\frac{\pi}{2})<0,$ so it has a solution.