Possible error in a proof in Jost's Riemannian Geometry and Geometric Analysis

You should know the following elementary properties of the codifferential $d^*$:

1. the adjoint relation $(d \psi, \omega) = (\psi, d^* \omega)$, often taken as the definition of $d^*$; and
2. the fact $d^* \circ d^* = 0,$ which follows from $d \circ d = 0$.

To show $d \psi = 0$ it suffices to show $(d \psi, \omega) = 0$ for an arbitrary 2-form $\omega$. From the first fact above we know $(d \psi, \omega) = (\psi, d^* \omega).$ The second fact tells us that $d^* (d^* \omega) = 0$, so $d^* \omega$ is in the kernel of $d^*$ and thus $(\psi, d^* \omega) = 0$ since we were given $\psi \perp \ker d^*.$