  Identities involving cyclic sums of terms composed from Jacobi elliptic
functions evaluated at $p$ equally shifted points on the real axis were
recently found. These identities played a crucial role in discovering linear
superposition solutions of a large number of important nonlinear equations. We
derive four master identities, from which the identities discussed earlier are
derivable as special cases. Master identities are also obtained which lead to
cyclic identities with alternating signs. We discuss an extension of our
results to pure imaginary and complex shifts as well as to the ratio of Jacobi
theta functions.
