sec(2x) - 4 = 0
sec(2x) = 4
1 / cos(2x) = 4
cos(2x) = 1/4
2x = cos⁻¹(1/4) + 2nπ or 2x = -cos⁻¹(1/4) + 2nπ
(where n is any integer)
x = 1/2 cos⁻¹(1/4) + nπ or x = -1/2 cos⁻¹(1/4) + nπ
The first two positive solutions occur for n = 0 in the first family of solutions, and for n = 1 in the second family, so
x = 1/2 cos⁻¹(1/4) or x = π - 1/2 cos⁻¹(1/4)
x ≈ 0.659 or x ≈ 2.483