What is the solution to arctan(i)

Question

I have no idea how to anywhere with this problem, can somebody please help me answer this question, I am a beginner at trig and am looking for some help

Answer 1

Suppose that $\tan z=i$. That means that $\sin z=i\cos z$. Then $$\sin^2 z+\cos^2 z=(i\cos z)^2+\cos^2z=(i^2+1)\cos^2 z=0.$$ But $$\sin^2 z+\cos^2 z=1.$$

Therefore there is no complex number with $\tan z=i$ (or with $\tan z=-i$).