I presume that exchanging the cosine by the sine will not matter for the large-$n$ behavior of the product, so let me consider $$a_n^2=4^n\prod_{k=1}^{n}\sin^2 k=\left(\prod_{k=1}^n\left|1-e^{k\alpha \pi i}\right|\right)^2\;\;\text{with}\;\;\alpha=2/\pi.$$ The convergence of $a_n$ was considered in this MO posting from five years ago, with lim inf $a_n$ equal to zero and lim sup $a_n$ equal to infinity for irrational $\alpha$.

I presume that exchanging the cosine by the sine will not matter for the large-$n$ behavior of the product, so let me consider $$a_n^2=4^n\prod_{k=1}^{n}\sin^2 k=\left(\prod_{k=1}^n\left|1-e^{k\alpha \pi i}\right|\right)^2\;\;\text{with}\;\;\alpha=2/\pi.$$ The convergence of $a_n$ was determined in this MO posting from five years ago, with lim inf $a_n$ equal to zero and lim sup $a_n$ equal to infinity for irrational $\alpha$.