Calculation of the input impedance of reactance modulator
Well $$Zin = \frac{Vx}{Ix} $$ 

where Vx is a input voltage and 

$$ Ix = \frac{Vx}{Z1+Z2} + gm*v $$ 
additional 
\$ v =Vx*\frac{Z2}{Z1+Z2}\$  

So we have this 

$$ Ix = \frac{Vx}{Z1+Z2} + \frac{gm*Vx*Z2}{Z1+Z2} $$ 

$$ Ix = Vx(\frac{1}{Z1+Z2} + \frac{gm*Z2}{Z1+Z2}) $$ 

$$ Ix = Vx(\frac{1+gm*Z2}{Z1+Z2}) $$ 

And finally 

$$ Zin = \frac{Vx}{Ix}=\frac{Z1+Z2}{1+gm*Z2} $$ 

Do you see now how we derive this expression?