Just observe that $\min_{0\le t\le 1}t+|2t^2-1|=1/\sqrt{2}$. That is, \[ \left|\cos 2^i x\right|+\left|\cos 2^{i+1}x\right|\ge\frac{1}{\sqrt{2}},\qquad\forall i\in\mathbb{N}_{\ge 0}. \]Apply this recursively to conclude the proof.