Calculus: Early Transcendentals, Chapter 4, 4.1, Section 4.1, Problem 40

You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation g'(theta) = 0 .
You need to evaluate the first derivative:
g'(theta) = 4 - 1/(cos^2 theta)
You need to solve for theta g'(theta) = 0, such that:
4 - 1/(cos^2 theta) = 0 => 4(cos^2 theta) - 1 = 0 => 4(cos^2 theta) = 1
(cos^2 theta) =1/4 => cos theta = +-1/2
cos theta = 1/2 => theta = +-arccos(1/2) + 2kpi
theta = +-pi/3 + 2kpi
cos theta = -1/2 => theta = pi+-arccos(1/2) + 2kpi
theta = pi + pi/3 + 2kpi => theta = (4pi)/3 + 2kpi
or
theta = pi - pi/3 + 2kpi => theta = (2pi)/3 + 2kpi
Hence, evaluating the critical numbers of the function for g'(theta) = 0, yields theta = +-pi/3 + 2kpi, theta = (4pi)/3 + 2kpi, theta = (2pi)/3 + 2kpi.