[tex]2+\cot A=1\implies \cot A=-1\implies \tan A=-1[/tex]
This happens whenever [tex]A=\dfrac{3\pi}4[/tex] or [tex]A=\dfrac{7\pi}4[/tex]. More generally, [tex]\tan A=-1[/tex] whenever you start with one of these angles and add any multiple of [tex]\pi[/tex], so the general solution would be [tex]A=\dfrac{3\pi}4+n\pi[/tex], where [tex]n[/tex] is any integer. (Notice that when [tex]n=1[/tex], you end up with [tex]\dfrac{7\pi}4[/tex].)
