A note on trigonometric identities involving non-commuting matrices

An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to converge in the convergent domain of the Zassenhaus formula and can be useful in the perturbative treatment of quantum mechanical problems, where exponentials of sums of non-commuting skew-Hermitian matrices frequently appear.