trig equation - all solutions

solve $\displaystyle cosec^{2}2x - cot2x = 1 $

$\displaystyle 0 \leq x \leq 180 $

replace $\displaystyle cosex^{2}2x $ with $\displaystyle 1 + cot^{2}2x = cosex^{2}2x $

$\displaystyle 1 + cot^{2}2x -cot2x = 1 $

$\displaystyle cot^{2}2x -cot2x = 0 $

$\displaystyle cot2x(cot2x-1) = 0 $

$\displaystyle cot2x = 0 $ or

$\displaystyle cot2x = 1 $

for $\displaystyle cot2x =0 $ is it correct to say there are no solutions for this?

for the other one I got $\displaystyle tan2x = 1 $

$\displaystyle 2x = 45, 225 $

$\displaystyle x = 22.5 , 112.5 $

Are these all the solutions in the interval, or does cot2x =0 also have a solution?