Confusion regarding calculation of fundamental period?
This seems like more of a semantics problem. 

A signal is periodic with time $T$ if

$$x(t+n\cdot T) = x(t), n \in \mathbb{Z}$$

So the signal is periodic in $0.5$ since the for $T = 0.5 \cdot n$ the argument of the cosine is an integer multiple of $2 \pi$. Since it's periodic in $0.5$ it's also periodic in all integer multiples of $0.5$, i.e $1$, $1.5$, $2$ etc. 

In this case it's also periodic in $0.25$ since 
$$ \cos^2(4 \cdot \pi \cdot t ) = 0.5 \cdot (1+\cos(8 \cdot \pi \cdot t))$$ 

So any periodic signal has an infinite number of periods, the fundamental one is the smallest one and all the others are integer multiples of the fundamental.