I want to prove $\exp(iAx) = I\cos x + iA\sin x$, where $I$ is the identity matrix $\in M_n(\mathbb{C})$, $A\in M_n(\mathbb{C})$ s.t. $A^2 = I$ and $A$ is normal, $x \in \mathbb{R}$, and $\exp(iAx)$ is defined in terms of the spectral decomposition of $A$. From $A^2 = I$ I know the moduli of the eigenvalues of $A$ are equal to 1, but I don't know how to go further. How do you do that?
EDIT: I clarified the implicit precondition on $A$ which is set in bold (but the proposition seems to hold without this). Also by "$\exp$ is defined in terms of the spectral decomposition", I mean this: let $\lambda_1,\dots,\lambda_n$ be eigenvalues of $A$ and $A=:\sum_{1\le k\le n}\lambda_k P_k$, where $P_k$ is the projection into the eigenspace that belongs to $\lambda_k$, and then define $\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$.