On one of Geronimus's results

In 1935, Ya. L. Geronimus found for the function $\sin(n+1)t-2q\sin{nt}$, $q\in\mathbb R$, the best integral approximation on the period $[-\pi,\pi)$ by the subspace of trigonometric polynomials of degree at most $n-1$. The result was an integral analog of the known theorem by E. I. Zolotarev (1868). At present, there are several methods of proving the mentioned fact. We propose one more variant of the proof. In the case $|q|\ge1$, we apply the $(2\pi/n)$-periodization as well as the orthogonality of the function $|\sin{nt}|$ and the harmonic $\cos t$ on the period. In the case $|q|<1$, we use the duality relations for P. L. Chebyshev's theorem (1859) on a rational function least deviating from zero on a segment in the uniform metric.