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)cos(π2−θ)=cos(π2)cos(θ)+sin(π2)sin(θ)
Use cos(pi/2) = 0cos(π2)=0 and sin(pi/2) = 1sin(π2)=1 to simplify:
cos(pi/2 - theta) = sin(theta)cos(π2−θ)=sin(θ)
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)