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: Tasks

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.