Which identity represents the secant function in terms of complementary angles?

The identity that represents the secant function in terms of complementary angles can be understood through the relationship between trigonometric functions and their complements. The secant function is defined as the reciprocal of the cosine function, meaning that sec(x) = 1/cos(x).

To explore the identity sec[(π/2) - x], first recognize that (π/2) - x is the complementary angle to x. In trigonometry, there are specific relationships between the trigonometric functions of an angle and its complementary angle.

For the secant of the complementary angle, we can derive:

sec[(π/2) - x] = 1/cos[(π/2) - x]. The cosine of a complementary angle relates to the sine function, specifically cos[(π/2) - x] = sin(x). Therefore, when we substitute this into the secant identity, we have:

sec[(π/2) - x] = 1/sin(x).

Recognizing that the reciprocal of the sine function is the cosecant function, we arrive at:

sec[(π/2) - x] = csc(x).

This identity is consistent with the fundamental properties of trigonometric identities involving complementary angles, confirming