How $x=1$ is integer solution of $\arctan \big(\frac{1}{\zeta(x)})=0$ and in the same time is not a solution of $\exp (-\zeta(x))=0$?

Really am mixed when I pluged the two functions in Wolfram alpha the first one show us that $$x=1$$ is integer solution of $$\arctan \big(\frac{1}{\zeta(x)})=0$$ and is not a solution for the second equation $$\exp (-\zeta(x))=0$$ , However both of exponential function and arctan function are continuous by means we can take $$\lim_{x\to 1 }\zeta(x) =+ \infty$$ and by substituion the two preceding equations are satisfied by $$x=1$$ , Now my question here is :

Question: How $$x=1$$ is integer solution of $$\arctan \big(\frac{1}{\zeta(x)})=0$$ and in the same time is not a solution of $$\exp (-\zeta(x))=0$$ ?

Learning $\arcsin, \arccos, \arctan$ – how to?

Sorry for asking such question.

I have a very basic understanding of $$\arcsin, \arccos, \arctan$$ functions. I do know how their graph looks like and not much more beyond that.

Calculate:

Which specific keywords should I google to learn how to solve the following tasks? I think those aren’t equations (googling ‘cyclometric equations’ was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?

Thank you.