[tex]\tan2B=\cot2B[/tex]
[tex]\tan2B-\cot2B=0[/tex]
[tex]\tan2B-\dfrac1{\tan2B}=0[/tex]
[tex]\dfrac{\tan^22B-1}{\tan2B}=0[/tex]
Note that we can't have [tex]\tan2B=0[/tex]. Meanwhile,
[tex]\tan^22B-1=0\implies\tan^22B=1\implies\tan2B=\pm1[/tex]
[tex]\tan2B=\pm1\implies2B=\pm\dfrac\pi4+n\pi[/tex]
where [tex]n[/tex] is any integer.
[tex]\implies\boxed{B=\pm\dfrac\pi8+\dfrac{n\pi}2}[/tex]
