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There are two functions:

1) f(x)=cos(nx)

2) f(x)=cos(x)

T=2π is the fundamental period of (2) function.

T1 is the fundamental period of (1) function.

How to prove that T1=2π/n?

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By definition, a function f(.) is periodic with period p if f(x+p) = f(x) and p is the smallest positive number that satisfies the above relation.

Hence cos(n(x+p))=cos(nx+np)=cos(nx)⟺np=2∗π.

Hence the period p=2π/n

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