I want to demonstrate the set :
$$\left\{(\cosh(x),\cosh(x)\tan(y),x),(x,y)\in \left]-3,3 \right[\times\bigcup_{k\in\mathbb{Z}}\left] \frac{-\pi}{2}+2k\pi,\frac{\pi}{2}+2k\pi \right[ \right\}$$ is a 2-dimensional submanifold of $\mathbb{R}^3$.
If it makes sense , how can i prove it ? Otherwise , why is it not a submanifold ?
I have thought about defining a mapping :
$\phi$ : $\left]-3,3 \right[\times\bigcup_{k\in\mathbb{Z}}\left] \frac{-\pi}{2}+2k\pi,\frac{\pi}{2}+2k\pi \right[ \rightarrow \mathbb{R}^3,(x,y)\rightarrow (\cosh(x),\cosh(x)\tan(y),x)$ and maybe try to play with the fact submanifolds of $\mathbb{R^n}$ are it's opened set , but there's no way this can properly define a homeomorphism i believe,
Update: Thanks to Didier for his contribution : The still unsolved problem can also be reformulated using the periodicity of tan function as demonstrate the set :
$$\left\{(\cosh(x),\cosh(x)\tan(y),x),(x,y)\in \left]-3,3 \right[\times\left] \frac{-\pi}{2},\frac{\pi}{2} \right[ \right\}$$ is a 2-dimensional submanifold of $\mathbb{R}^3$ if this makes sense.