Start with the identity for the cosine of the difference of two angles:
#cos(pi/2 - theta) = cos(pi/2)cos(theta) + sin(pi/2)sin(theta)#
Use #cos(pi/2) = 0# and #sin(pi/2) = 1# to simplify:
#cos(pi/2 - theta) = sin(theta)#
Substitute #sqrt(1 - cos²(theta))# for #sin(theta)#:
#cos(pi/2 - theta) = sqrt(1 - cos²(theta))#
Substitute #1/6# for #cos(theta)#:
#cos(pi/2 - theta) = sqrt(1 - (1/6)²)#
#cos(pi/2 - theta) = sqrt(35)/6#
Use the identity for the sine of the difference of two angles:
#sin(pi/2 - theta) = sin(pi/2)cos(theta) - cos(pi/2)sin(theta)#
Use #cos(pi/2) = 0# and #sin(pi/2) = 1# to simplify:
#sin(pi/2 - theta) = cos(theta)#
Substitute #1/6# for #cos(theta)#:
#sin(pi/2 - theta) = 1/6#
Use #cot(x) = cos(x)/sin(x)#:
#cot(pi/2 - theta) = cos(pi/2 - theta)/sin(pi/2 - theta)#
Substitute in the values from above:
#cot(pi/2 - theta) = (sqrt(35)/6)/(1/6)#
#cot(pi/2 - theta) = sqrt(35)#
