= Solution
A <photon> of <wavelength> $\lambda$ has <energy> $E=h\nu=hc/\lambda$ and <momentum> $\mathbf p=(h/\lambda)\hat{\mathbf n}$ along its propagation direction.
The specified <Compton scattering> formula requires the electron to be initially at rest in the frame where the wavelengths and angle are measured. This initial-rest assumption is implicit in the printed request; it is not valid for arbitrary initial electron <momentum>. Use <four-momentum> with <Minkowski metric> signature $(+---)$: initially $P=(mc,\mathbf0)$, while $K=(E/c,(E/c)\hat{\mathbf n})$ and $K'=(E'/c,(E'/c)\hat{\mathbf n}')$ are null. Conservation gives $P'=P+K-K'$, and the outgoing electron satisfies the same <mass-shell condition> $P'^2=P^2=m^2c^2$. Hence
$$
P\cdot(K-K')=K\cdot K',\qquad m(E-E')=\frac{EE'}{c^2}(1-\cos\theta).
$$
After dividing by $EE'$ and substituting the <photon> <energy> relation,
$$
\frac1{E'}-\frac1E=\frac{1-\cos\theta}{mc^2},\qquad \boxed{\lambda'-\lambda=\frac h{mc}(1-\cos\theta).}
$$
The scale $h/(mc)$ is the electron <Compton wavelength>. For a moving initial electron, the earlier invariant identity remains valid, but $P\cdot K$ includes its initial velocity and the wavelength shift is different.
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