A spectral product formula for repunits via a tridiagonal Toeplitz similarity

For $b>0$ and $n⩾1$, we consider the $n\times n$ tridiagonal matrix $V_n(b)$ with diagonal entries $b+1$, superdiagonal entries $1$, and subdiagonal entries $b$. A diagonal similarity reduces $V_n(b)$ to a symmetric tridiagonal Toeplitz matrix and hence makes its spectrum explicit. Since $\det (V_n(b))$ equals the geometric sum $1+b+\cdots +b^n$, taking determinants yields a finite cosine product evaluation for this quantity. As further consequences, we derive sharp bounds from the extremal eigenvalues, write down explicit eigenvectors with respect to a natural weighted inner product, and obtain a closed formula for $V_n(b{)}^{-1}$.