I am hoping to find whether the following holds or not:
$\dfrac{\phi(-x)}{1-\Phi(-x)}$ = $\dfrac{\phi(x)}{\Phi(x)}$
I am working through Econometric Analysis (Greene) and manage to reach the expression on the right-hand side. However, the final solution in the book gives the one on the left-hand side.
If it does, then please state the source as well.
This identity follows from the following: Since standard normal distribution is symmetric about $0$ we have $\Phi (x)=1-\Phi(-x)$. In other words $\frac {\Phi (x)} {1-\Phi(-x)}=1$. Differentiate this using quotient rule to derive your identity.
As observed by Math1000 in his comment below we can simply use the fact that $\Phi (x)= 1-\Phi(-x)=1$ together with the fact that $\phi$ is an even function to complete the proof.