The angle #alpha=-(3pi)/2# means the point #(0,1)# on the unit circumference (center #(0,0)# and radius #1#). In fact, #pi/2# is a quarter of a turn, and you must do three quarters clockwise starting from #(1,0)#.
That being said, remember that #cos(alpha)# and #sin(alpha)# are, respectively, nothing but the #x# and #y# components of any point laying on the circumference. We easily have #cos(-(3pi)/2) = 0#, and #sin(-(3pi)/2) = 1#.
Knowing these values, the others are straightforward:
#tan(-(3pi)/2) = \frac{sin(-(3pi)/2)}{cos(-(3pi)/2)}# is undefined.
#cot(-(3pi)/2) = \frac{cos(-(3pi)/2)}{sin(-(3pi)/2)} = 0/1 = 0#
#csc(-(3pi)/2) = 1/(sin(-(3pi)/2)) = 1/1 = 1#
#sec(-(3pi)/2) = 1/(cos(-(3pi)/2))# is undefined.
