Answer
True.
Work Step by Step
If $c$ is the period of $f(t)$, then the period of $f(\frac{1}{2}t)$ is $2c$
$f(\frac{1}{2}[t+4c])=f(\frac{1}{2}t)$, because $4c$ is a multiple of $2c$, that is, $4c=2(2c)$
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