Use the identity #cot(theta) = 1/tan(theta)#:
#cot(theta) = -12/5#
Use the identity:
#1+tan^2(theta) = sec^2(theta)#
Substitute #tan^2(theta) = (-5/12)^2 #:
#1+(-5/12)^2 = sec^2(theta)#
#144/144+25/144#
#169/144 = sec^2(theta)#
#sec(theta) = +-13/12#
We know that the secant is negative in the second quadrant, therefore, we choose the negative value:
#sec(theta) = -13/12#
Use the identity #cos(theta) = 1/sec(theta)#:
#cos(theta) = -12/13#
Use the identity:
#tan(theta)= sin(theta)/cos(theta)#
Multiply both sides by #cos(theta)#:
#sin(theta) = tan(theta)cos(theta)#
Substitute #tan(theta) = -5/12 and cos(theta) = -12/13#:
#sin(theta) = (-5/12)(-12/13)#
#sin(theta) = 5/13#
Use the identity #csc(theta) = 1/sin(theta)#:
#csc(theta) = 13/5#
