Projector phase rotation ($e^{-i\theta P}=I+(e^{-i\theta }-1)P$)

For an orthogonal projection, $P^k=P$ for $k\ge 1$. Summing the exponential series proves the displayed identity. At $\theta =\pi$, the operator is $I-2P$, which changes the sign on the range and fixes its orthogonal complement. In Grover search algorithm, the marked-state reflection and the uniform-state reflection compose to the search iterate up to an irrelevant global phase.