The limit is $0$; we will show this by checking that $d_n = 0$ for all $n \geq 3$.
Starting from the given matrix, add the third column to the first column; this does not change the
determinant. However, thanks to the identity
$\cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$,
the resulting matrix has the form
$$\begin{pmatrix} 2 \cos 2 \cos 1 & \cos 2 & \cdots \\ 2 \cos (n+2) \cos 1 & \cos (n+2) & \cdots \\ 2 \cos (2n+2) \cos 1 & 2 \cos (2n+2) & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}$$
with the first column being a multiple of the second. Hence $d_n = 0$.