Question 789117
{{{csc(-13pi/3)=csc(-12pi/3-pi/3)=csc(-4pi-pi/3)=csc(-pi/3)}}}
That is because {{{-13pi/3=-12pi/3-pi/3-4pi-pi/3}}}
The added {{{-4pi}}} angle is 2 clockwise turns, so the {{{-pi/3}}} and {{{-4pi-pi/3}}} angles are coterminal angles, indistinguishable from each other unless you are watching the turns.
(Shaun White may say there is a big difference between a {{-60^o}}} and a {{{-720^o-60^o=-780^o}}}, but the turns end in the same position, and the values of the trigonometric functions are the same.
 
Next {{{csc(-pi/3)=1/sin(-pi/3)}}}
and {{{sin(-pi/3)=-sin(pi/3)=-sqrt(3)/2}}}
So {{{csc(-13pi/3)=1/(-sqrt(3)/2)=-2/sqrt(3)=-2sqrt(3)/2}}}