$L$-approximation of a linear combination of the Poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination $\Pi_{q,\alpha}=\cos(\alpha\pi/2)P+\sin(\alpha\pi/2)Q$ of the Poisson kernel $P(t)=1/2+q\cos t+q^2\cos2t+\dots$ and its conjugate kernel $Q(t)=q\sin t+q^2\sin2t+\dots$ is considered for $\alpha\in\mathbb R$ and $|q|<1$. A new explicit formula is found for the value $E_{n-1}(\Pi_{q,\alpha})$ of the best approximation in the space $L=L_{2\pi}$ of the function $\Pi_{q,\alpha}$ by the subspace of trigonometric polynomials of order at most $n-1$. Namely, it is shown that
$$E_{n-1}(\Pi_{q,\alpha})=\frac{|q|^n(1-q^2)}{1-q^{4n}}\left\|\frac{\cos(nt-\alpha\pi/2)-q^{2n}\cos(nt+\alpha\pi/2)}{1+q^2-2q\cos t}\right\|_L.$$
Besides, the value $E_{n-1}(\Pi_{q,\alpha})$ is represented as a rapidly converging series.