Answer
Please see the work below.
Work Step by Step
We know that for a given length of exposure, the amount of light falling on the film is directly proportional to the area of the aperture. Thus, brightness $\propto D^2$
and we also know that $D=\frac{f}{f-number}$
Now $brightness \propto (\frac{f}{f-number})^2$. Hence it is proved that the brightness is inversely proportional to the square of the $f-number$.
