The linear wave equation on $\mathbb{R}^d$ is given by $\Box u = 0, (u|_{t = 0}, \partial_t u|_{t = 0}) = (u_0, u_1)$, where $\Box = \partial_t^2 + \Delta$. This can be solved by "factoring" the d'Alembertian as $\Box = (i\partial_t + |\nabla|)(i\partial_t - |\nabla|)$, setting $U(t, x) = i\partial_t u - |\nabla| u$, and proceeding to find that $$ U(t, x) = -|\nabla|[\cos(t|\nabla|)u_0 + \frac{\sin(t|\nabla|)}{|\nabla|}u_1] + i[-|\nabla|\sin(t|\nabla|)u_0 + \cos(t|\nabla|)u_1]. $$ From this, the solutions I have seen then assert that (by definition of $U$) that the solution $u$ is $$ u(t, x) = \cos(t|\nabla|)u_0 + \frac{\sin(t|\nabla|)}{|\nabla|}u_1. $$ The argument is that the second brackets term is $\partial_t$ of the first brackets term, and using the form of $U$, they conclude that $u$ must in fact be the term in the first square brackets above.
My question: How is this justified? If maybe every function here was real-valued, I could believe it (since we then just need to match real parts with real parts, and imaginary with imaginary), but I can't see that this is the case here. Is it perhaps an assumption that, without loss of generality, we can take $u$ to be real-valued? Or is it a different argument?
Any help is much appreciated!