Some trigonometric identities related to exact covers

Abstract:

Sherman K. Stein proves that if $\sin \pi z = k\prod \limits _{i = 1}^n {\sin } \left ( {\pi /{d_i}} \right )\left ( {{b_i} - z} \right )$ where the ${b_i}$ are integers, the ${d_i}$ are positive integers, $k$ is a constant, then $\left \{ {\left ( {{d_i}:{b_i}} \right )} \right \}$ is an exact cover. It is shown here that if $0 \leq {b_i} < {d_i}$ then $k = - {2^{n - 1}}$, that the converse is also true, and an analogous formula is conjectured for infinite exact covers. Many well known and lesser known trigonometric and functional identities can be derived from this result and known families of exact covers. A procedure is given for constructing exact covers by induction.