Given: #tan(x) = 5/12# and x is acute.
Starting with the identity:
#sin^2(x) + cos^2(x) = 1#
Divide both sides by cos^2(x):
#sin^2(x)/cos^2(x) + 1 = 1/cos^2(x)#
Use the identity #sin(x)/cos(x) = tan(x)#:
#tan^2(x) + 1 = 1/cos^2(x)#
Use the identity #1/cos(x) = sec(x)#
#tan^2(x) + 1 = sec^2(x)#
Substitute #(5/12)^2# for #tan^2(x)#
#(5/12)^2 + 1 = sec^2(x)#
#25/144 + 1 = sec^2(x)#
#25/144 + 144/144 = sec^2(x)#
#169/144 = sec^2(x)#
#sec(x) = +-13/12#
But we drop the negative, because we are told that x is acute:
#sec(x) = 13/12#
#sec(x) = 1/cos(x) #
#cos(x) = 12/13#
#sin(x) = tan(x)cos(x)#
#sin(x) = 5/12 12/13#
#sin(x) = 5/13#
#csc(x) = 1/sin(x)#
#csc(x) = 13/5#
#cot(x) = 1/tan(x)#
#cot(x) = 12/5#
