$(10$ pts$)$ In $1923,$ Compton demonstrated the particle nature of light through experiments. When Xrays with an energy $E_0$ are incident on carbon $T,$ electrons are scattered at an angle $$ with the xaxis, and the scattered X $-$rays are measured at an angle $.$

$($a$)(5$pt$)$ Show that the kinetic energy of recoil the electron is

$\frac{E_k}{E_0}=\frac{\frac{E_0}{m_e{c}^2}(1-cos)}{1+\frac{E_0}{m_e{c}^2}(1-cos)}$

$($b$)(5$ pts$)$ Show that the recoil angle $$ of the electron is

$cot=(1+\frac{E_0}{m_e{c}^2})tan(\frac{}{2})$