Check proof that $\tan(x+\sin(x))$ is periodic and find its period

Question

I need to:

Show that $\tan(x+\sin(x))$ is periodic and find its period

I've tried to do that by definition of periodic functions, so:

$$ \tan(x + \sin(x)) = \\ = \tan(x+T+ \sin(x+T)) = \\ =\tan(x+T+\sin(x)\cos(T)+\cos(x)\sin(T)) $$

From the above i see that $\cos(T) = 1$ and $\sin(T) = 0$ so $T= 2\pi n$. Let's check the function for $T=2\pi n$:

$$ \tan(x+\sin(x)) = \tan(x+2\pi n + \sin(x)\cos(2\pi n) + \cos(x)\sin(2\pi n)) $$

Let $n = 1$ for $T$ to not be equal to $0$:

$$ \tan(x+\sin(x)) = \tan(x+2\pi + \sin(x)\cos(2\pi) + \cos(x)\sin(2\pi)) = \tan(x + \sin(x) + 2\pi) $$

But $\tan(2\pi + \alpha) = \tan(\alpha)$, so the above equation holds for $T=2\pi n$. Basically I have 2 questions:

1. Is the reasoning above correct?
2. I'm in doubt because W|A query says the function is not periodic, who's right?

Answer 1

Yes, the function is periodic. Your approach complicates things a lot. All you have to do is to note that $\tan\bigl(x+2\pi+\sin(x+2\pi)\bigr)=\tan\bigl(x+\sin(x)\bigr)$.