Given: #cos(x) = sqrt(3)/2#
Use the identity #sin(x) = +-sqrt(1 - cos^2(x))#:
Because we are given that #tan(x)# is a positive number, we shall drop the #+-#, thereby, making the sine positive, only:
#sin(x) = sqrt(1 - (sqrt(3)/2)^2)#
#sin(x) = sqrt(4/4 - 3/4)#
#sin(x) = 1/2#
Verify that #tan(x) = sqrt(3)/3#:
#sin(x)/cos(x) = (1/2)/(sqrt(3)/2) = 1/sqrt(3) = sqrt(3)/3 = tan(x)#
Verified.
Use the identity #cot(x) = 1/tan(x)#:
= 1/(sqrt(3)/3)#
#cot(x) = sqrt(3)#
Use the identity #csc(x) = 1/sin(x)#
#csc(x) = 1/(1/2)#
#csc(x) = 2#
Use the identity #sec(x) = 1/cos(x)#
#sec(x) = 1/(sqrt(3)/2)#
#sec(x) = 2/sqrt(3)#
#sec(x) = (2sqrt(3))/3#
