Can someone help me prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & -m \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} 2 & \text{if } n=1\pmod{2}\\ 2m^{\frac{n}{2}} & \text{if } n=0\pmod{4}\\ -2m^{\frac{n}{2}}-4 & \text{if } n=2\pmod{4}\\ \end{cases}, $$

For the 1st equality, since the product of $2\cos\frac{2j\pi}{n}$'s is 2 when $n$ is odd, we only need to prove that the trace is a constant polynomial in $m$. However, because of the cosine term, the approach of polynomial analysis given in [my previous post] [1] does not seem to work here.

Situations are the same for the 2nd and 3rd equalities. [1]:Product of a Finite Number of Matrices Related to Roots of Unity

Can someone help me prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & -m \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} 2 & \text{if } n=1\pmod{2}\\ 2m^{\frac{n}{2}} & \text{if } n=0\pmod{4}\\ -2m^{\frac{n}{2}}-4 & \text{if } n=2\pmod{4}\\ \end{cases}, $$

For the 1st equality, since the product of $2\cos\frac{2j\pi}{n}$'s is 2 when $n$ is odd, we only need to prove that the trace is a constant polynomial in $m$. However, because of the cosine term, the approach of polynomial analysis given in [my previous post] [1] does not seem to work here.

Situations are the same for the 2nd and 3rd equalities. [1]:Product of a Finite Number of Matrices Related to Roots of Unity