sec2x

sec2x denotes the secant of the double angle, written as sec(2x). It is defined for all x where cos(2x) ≠ 0 and equals 1 divided by cos(2x).

The domain of sec(2x) is all real x except those where cos(2x) = 0, which occur at x = π/4 + k·π/2 for integers k. Its range is y ≤ −1 or y ≥ 1. The function has period π, since cos(2x) has period π. It is an even function, satisfying sec(−2x) = sec(2x).

Sec(2x) has vertical asymptotes at the points x = π/4 + k·π/2, where cos(2x) = 0. Between asymptotes, the graph consists of alternating branches that approach the asymptotes from above or below, reflecting the sign of cos(2x).

Related identities include sec(2x) = 1/cos(2x) and cos(2x) = cos^2 x − sin^2 x, so sec(2x) = 1/(cos^2 x − sin^2 x). The double-angle form connects sec(2x) to other trigonometric functions via standard sum-to-product and tangent double-angle relationships.

Calculus notes: the derivative of sec(2x) is d/dx [sec(2x)] = 2 sec(2x) tan(2x). An antiderivative is ∫ sec(2x) dx = (1/2) ln|sec(2x) + tan(2x)| + C, using the substitution u = 2x.