 Similarly, as the Moyal bracket equals the Poisson bracket up to higher orders of , the cosine bracket equals the ordinary product up to higher orders of .In the classical limit, the Moyal bracket helps reduction to the Liouville equation (formulated in terms of the Poisson bracket), as the cosine bracket leads to the classical Hamilton–Jacobi equation.The sine and cosine bracket also stand in relation to equations of a purely algebraic description of quantum mechanics.