Correct graph of arccot x

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There are 2 possible ways of choosing an interval of cot x graph for obtaining the graph of arccot x. Why do we choose the interval (0,$\pi$) instead of the first choice? This may be because the graph is continous at that interval,but the graphs of arcsec x and arccosec x aren't continous too and we don't generally face problems working with them, so what is the reason in this case? pa enter image description here