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It is graphically evident and easy to prove that if for $\theta\in\mathbb R$ one has $\cos^2\theta \ge c:=\cos^2\frac12=0,7701\dotso<1$, then $|\cos^2(\theta+1)|\le c$. As a consequence the number of integers $k\in\{0,\dotsc,n-1\}$ such that $\cos^2 k\ge c $ is at most $n/2$ (in fact it is even larger, due to the uniform distribution of $e^{2ki}$ on the circle). This ensures the product for $c_n$ an exponential decay, so that $nc_n=o(1)$.