Let $φ:E→E$ be $q$-th Frobenius map and $φ'$ be it's dual isogeny. And we define $a$ as $a＝1-deg(1-φ)＋deg(φ)$(what we call trace of frobenius).
Then, My question is ,
Why $[a]＝φ＋φ'$ ?
These questions are from Silverman's 'the arithmetic of elliptic curves', $p150$. In the process of counting the number of supersingular elliptic curve over closed field of positive character, I need to use this result.