How do I prove that $P_pm=frac12(1pm U)$ if $U^2=I$?

First, we can start with $U = P_+ - P_-$, since the Hermitian is the sum of the projection operators of the eigenspaces scaled by their eigenvalues. If $U^2 = I$, that means $I = (P_+ - P_-)(P_+ - P_-) = P_+P_+ - P_+P_- - P_-P_+ + P_-P_-$. Since the two projectors correspond to orthogonal eigenspaces, operating on themselves doesn't change them and one operating on the other gets a zero, so we get $P_+ + P_- = I$. With $P_+ + P_- = I$ and $P_+ - P_- = U$, we can add the two equations to get $P_+ = frac{1}{2}(I + U)$ or subtract the $U$ from the $I$ to get $P_- = frac{1}{2}(I - U)$.