Recall the half-angle identity,
cos²(x/2) = (1 + cos(x))/2
The angle A terminates in the fourth quadrant, so 270° < A < 360°. Then the corresponding half-angle terminates in the second quadrant, since 135° < A/2 < 180°, so cos(A/2) < 0. This means
cos(A/2) = - √((1 + cos(A))/2) = -3/4
so that
sec(A/2) = 1/cos(A/2) = -4/3.
