Period of $\sin^{2} x\ + \cos^{2} x$

Question

While I was studying Periodicity of functions I encountered the following problem :

My textbook said that :

Constant function is a periodic function but its fundamental period doesn't exist

So to check it I assumed $$f(x) = \sin^{2}x + \cos^{2} x$$ $$f(x) = 1$$ Now if I go by definition i.e.

If $f(x+T) = f(x)$ for some value of T then the smallest value for T is known as fundamental period of $f(x)$

by definition it comes out to be $\frac \pi 2$ but if we go by the constant function it should not exist . So I am confused to what to choose : $\frac \pi 2$ or doesn't exist ?

Answer 1

The given function $f(x) = \sin^2 x + \cos^2 x$ simplifies to the constant function $f(x) = 1$. This is because of the trigonometric identity $\sin^2 x + \cos^2 x = 1$, which holds for all $x$.

Now, regarding periodicity, a function is called periodic if there exists a non-zero value $T$ such that $f(x + T) = f(x)$ for all $x$. For a constant function, this condition is satisfied for any non-zero value of $T$, as $f(x) = f(x + T)$ for any $x$ and any $T$.

The fundamental period of a function is the smallest positive value of $T$ that satisfies the periodicity condition. However, in the case of a constant function, since any non-zero value of $T$ satisfies the periodicity condition, there is no smallest positive value of $T$. Therefore, the fundamental period of a constant function does not exist.

In your example, the fundamental period of $f(x) = \sin^2 x + \cos^2 x$ does not exist, as it is a constant function. The confusion with $\frac{\pi}{2}$ arises from considering the individual functions $\sin^2 x$ and $\cos^2 x$, which are periodic with period $\pi$. However, their sum, which is the constant function $f(x) = 1$, does not have a fundamental period.

Answer 2

$\dfrac{\pi}{2}$ is not fundamental period of $f$ since we have $0<\dfrac{\pi}{3}<\dfrac{\pi}{2}$ and $f(x+\tfrac{\pi}{3})=f(x)$.

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Your textbook says, "Constant function is a periodic function but its fundamental period doesn't exist."

According to that, fundamental period of $f$ doesn't exist because $f$ is a constant function.

Answer 3

$\frac{\pi}{2}$ is a period of $\sin^2x+\cos^2x$

Even $e^{\pi^e}$ is a period of $\sin^2x+\cos^2x$

You get it$?$ Any real number (except $0$) can be the period of $\sin^2x+\cos^2x$