Josef Wichmann's harmonic Wronskians (Q1824798)

Attention is drawn to the beautiful formula for the Wronskian of sines: \[ \left| \begin{matrix} \sin x,\sin 2x,...,\sin nx\\ \text{``derivatives''} \end{matrix} \right| =(-2)^{(n(n-1)/2)}(\sin x)^{(n(n+1)/2)}\prod^{n-1}_{j=0}j!. \] To satisfy the natural curiosity concerning the accompanion of cosines, it is computed to \[ 2^{(n(n-3)/2)}(-\sin x)^{(n(n-1)/2)}\prod^{n- 1}_{j=0}j!\sum^{[n/2]}_{k=0}\binom{n}{2k} \binom{2k}{k}(2 \cos x)^{n-2k}. \] This result underlines a difference between sine and cosine; so the proofs rely not surprisingly on the Chebyshev polynomials.