For each real number $a$ with $0\le a\le 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0,a]$ and $[0,1]$, respectively, and let $P(a)$ be the probability that

${\cos }^2(\pi x)+{\cos }^2(\pi y)>1$.

What is the maximum value of $P(a)$?