Integer multiples of $2\pi$ on $\cos$ function

Integer multiples of $2\pi$ on $\cos$ function

Suppose you know that the smallest positive value $t$ such that $\cos t=0$
is $t=\dfrac{\pi}{2}$, but you don't know other values of $\cos$ and
$\sin$.
You also know that $u$ is a real number such that $\cos u=1$ and $\sin u=0$.
Just from this, can you conclude that $u$ must be an integer multiple of
$2\pi$?
(You know the usual trig identities, such as $\cos^2x+\sin^2x=1$ and $\cos
2x=2\cos^2x-1$.)