Let $\hat{\mathbf{x}}$ be the direction of the incoming photon, and let $\hat{\mathbf{y}}$ be the direction of the outgoing photon. Let $E'$ be the energy of the photon after it is scattered and let $(p_x,p_y)$ be the momentum of the electron after the the photon scatters off of it. Conservation of energy and momentum gives
$\begin{align*}E + mc^2 &= E' + \sqrt{p_x^2 c^2 +p_y^2 c^2 + m^2 c^4} \\\frac{E}{c} &= p_x \\0 &= \frac{E'}{c} + p...$
Plugging the second and third equations into the first equation gives
$\begin{align*}&& E + mc^2 &= E' + \sqrt{E^2 + E^2 + m^2 c^4} \\&\Longrightarrow& E - E' + mc^2 &= \sq...$
Therefore, answer (E) is correct.

Solution to 2008 Problem 97