Using the spin operator commutation relations, we see that the commutators evaluate to i S y for the odd terms in the series, and to S x for all of the even terms. Thus: U † † S x U = S x [ 1 − − θ θ 2 2 ! + ⋯ ⋯ ] − − S y [ θ θ − − θ θ 3 3 ! ⋯ ⋯ ] = S x cos ⁡ ⁡ θ θ − − S y sin ⁡ ⁡ θ θ , as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum number s)