7. $2 \pi \cdot f(x+T)=f(x) \Leftrightarrow \cos ^{7}(x+T)=\cos ^{7} x \Leftrightarrow \cos (x+T)=\cos x$, so the given function has the same smallest positive period as $y=\cos x$, which is $2 \pi$.
7. The smallest positive period of the function $f(x)=\cos ^{7} x$ is

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