Start by letting arcsin(sqrt(5)/6) to be a certain angle alpha
It follows that alpha=arcsin(sqrt5/6)
and so
sin(alpha)=sqrt5/6
This means that we are now looking for cot(alpha)
Recall that : cot(alpha)=1/tan(alpha)=1/(sin(alpha)/cos(alpha))=cos(alpha)/sin(alpha)
Now, use the identity cos^2(alpha)+sin^2(alpha)=1 to obtain cos(alpha)=sqrt((1-sin^2(alpha)))
=>cot(alpha)=cos(alpha)/sin(alpha)=sqrt((1-sin^2(alpha)))/sin(alpha)=sqrt((1-sin^2(alpha))/sin^2(alpha))=sqrt(1/sin^2(alpha)-1)
Next, substitute sin(alpha)=sqrt5/6 inside cot(alpha)
=>cot(alpha)=sqrt(1/(sqrt5/6)^2-1)=sqrt(36/5-1)=sqrt(31/5)=color(blue)(sqrt(155)/5)
