Why we can't find inverse function

Question

I've found that it isn't because it isn't an injection. But i can't find why is that... Can somebody explain this to my step by step?

$$ y= \sqrt[3]{\frac{\sqrt[5]{\sin x}-2}{\sqrt[5]{\sin x}+2}}$$

Answer 1

Note that, in particular, it is $$y(0)=y(2\pi)=-1.$$ But, since $\sin(x+2\pi)=\sin x$ it is $$y(x+2\pi)=y(x),\forall x\in\mathbb{R}.$$ That is, $y$ is periodic with period $2\pi.$ And a periodic function has not inverse.