There is a U.S. presidential election every 20 years. For example, there were elections in
1880,
1900,
1920,
1940,
1960,
1980, and
2000. So the U.S. elections have a period of 20 years; they recur every 20 years.
But also, U.S. elections recur every four years
(for example,
1980,
1984,
1988,
1992,
1996, and
2000)
and their 20-year recurrence is a simple result of their recurrence every four years; if something happens every four years, like the elections, then it must also happen every twenty years, like the elections.
The elections have a period of twenty years, because they recur every twenty years. But they have a fundamental period of four years, because they recur every four years, and not any smaller amount.
Each function's period obviously divides $2\pi$. Let's take it from there.
As Michal Zapala has noted, $f=\sqrt{2}\sin(x+\frac{\pi}{4})$ has period $2\pi$.
Since $g=\tan x\,(1+\cos x)$ cannot return to its $x=0$ value of $0$ until $\tan x=0$ at multiples of $\pi$ or $\cos x =-1$ at odd multiples of $\pi$, $\pi$ divides the period, so it's $\pi$ or $2\pi$. But $x\mapsto x+\pi$ changes the sign of $g$, so the period will have to be $2\pi$ after all.
Similarly, since $h(x)=h(0)\implies \tan x = 0\implies \pi | x$, $h$ has period $\pi$ or $2\pi$. In fact this time the period is $\pi$, since $h(x+\pi)=h(x)$.
Best Answer
I ended up realizing that the easiest way to go about it would be to convert the $ cos^2(2\pi t) $ into a term without a square via the double angle formulas, so that it would turn into $ 1 \over 2 $ * $ cos(4\pi t) $ and some constant. Using the angular frequency formula above, the period then is 0.5s.