Alternate Method
I think the question is using known trigonometric identities.
As pi/3*1/3=pi/9 we can compute sec (pi/9) using cos (pi/3).
The method is as given below
cos (pi/3) = 1/2
cos(3 pi/9) = 1/2
cos 3theta = cos(2theta +theta)
= cos2thetacostheta-sin2thetasintheta
=2cos^2thetacostheta-costheta-2sin^2thetacostheta
=4cos^3theta-3costheta
If theta = pi/9 and we assume x=costheta then we have
4x^3-3x-1/2 = 0
8x^3-6x-1 = 0
Solving the equation results in 3 roots.
x=costheta=( -0.76604, -0.17365, 0.93969) as the angle is less than pi/2 cosine is positive. Hence choose the positive root.
cos(pi/9) = 0.93969
sec (pi/9) = 1/cos(pi/9) = 1/0.93969
sec(pi/9) = 1.0642
