I was hoping that someone could $(1)$ verify for me that this proof is correct (or let me know where I messed up), and $(2)$ answer my question that is after the proof. Thanks in advance for your time.
- Thank you to @MichaelAlbanese for helping me with this proof.
(1) Your proof looks fine.
(2) Yes, to refute (disprove) an "always" statement a single counterexample is sufficient. In this case, providing $x=\pi/2$ as a counterexample works, precisely because it demonstrates that the domains of the two functions are not the same: it belongs to the domain of one of them ($\cot\theta$) but not of the other. Thus the domains are not the same, and therefore the functions are not the same.
Of course, we can say that $\cot\theta=\frac{1}{\tan\theta}$ whenever both sides are defined. But as your proof shows, this extra clause "whenever both sides are defined" is needed because, technically speaking, they are not the same functions if considered on their natural domains.