Question 833905
You probably meant that {{{cot(theta)=1/tan(theta)}}} <--> {{{tan(theta)=1/cot(theta)}}}, 
so applying that to your problem,
{{{ cot(x/3)= sqrt( 3 ) }}} --> {{{tan(x/3)= 1/sqrt( 3 ) }}}.
It is true that {{{tan(30^o)=1/sqrt(3)}}} and that {{{arctan(1/sqrt(3))=30^o}}} .
It is also true that {{{tan(30^o+180^o)=tan(210^o)=1/sqrt(3)}}} ,
and that, generalizing, {{{tan(30^o+n*180^o)=1/sqrt(3)}}} for any integer {{{n}}} .
{{{x/3=30^o}}} means {{{x=3*30^o}}} so {{{highlight(x=90^o)}}} is a solution,
and it is the only one with {{{x}}} in [0,360) in degrees.
The other possible angle values do not lead to solutions with {{{x}}} in [0,360):
{{{x/3>=210^o}}} --> {{{x>=3*210^o}}} --> {{{x>=630^o}}} , and
{{{x<=30^o-180^o}}} --> {{{x<=-150^o}}} --> {{{x<=3*(-150^o)}}} --> {{{x<=-450^o}}}