It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1\}$ satisfy that $k\bmod\pi$ lies in $[\pi/3,2\pi/3]$. It follows, for $n$ large, that $\prod_{k=1}^{n-1}\cos^2(k)\leq 2^{-n/2}$. Hence the limit in the original question equals zero.

It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1\}$ satisfy that $k\bmod\pi$ lies in $[\pi/3,2\pi/3]$. It follows, for $n$ large, that $\prod_{k=1}^{n-1}\cos^2(k)\leq 2^{-n/2}$. Hence the limit in the original question equals zero.

Remark. From $\int_0^\pi\ln(\cos^2(x))\,dx=-\pi\ln 4$ it even follows that $\prod_{k=1}^{n-1}\cos^2(k)\leq 4^{-(1+o(1))n}$.

It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the positive integers $k\in\{1,\dotsc,n-1\}$ are such that $k\bmod\pi$ lies in $[\pi/3,2\pi/3]$. It follows, for $n$ large, that $\prod_{k=1}^{n-1}\cos^2(k)\leq 2^{-n/2}$. Hence the limit in the original question equals zero.

It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1\}$ satisfy that $k\bmod\pi$ lies in $[\pi/3,2\pi/3]$. It follows, for $n$ large, that $\prod_{k=1}^{n-1}\cos^2(k)\leq 2^{-n/2}$. Hence the limit in the original question equals zero.