# Is f(x) = cos^2(x) + sin^2(x) a periodic function?

## Homework Statement

Is f(x) = cos^2(x) + sin^2(x) a periodic function?

## Homework Equations

sin^2(x) + cos^2(x) = 1

## The Attempt at a Solution

This question is just something that randomly came to my mind (not a hw problem). I know cos^2(x) and sin^2(x) are both periodic functions, but is sin^2(x) + cos^2(x) a periodic function too? If so, what would be its fundamental frequency?

STEMucator
Homework Helper
No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.

No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.

Indeed, as Zondrina mentioned, it is not periodic. Typically, we say a function f(x) is periodic if there is a smallest positive integer P≠0 for which f(x+P)=f(x). The number P is then the period. Since sin^2(x)+cos^2(x)=1:=1(x), it is not periodic, because 1(x+P)=1=1(x) for any P (so in particular, there isn't a smallest one).

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Technically a constant function is periodic but has no fundamental period, but it is kind of silly to refer to it in such a way.