Reciprocal Trigonometric Functions

The reciprocal trigonometric functions — secant (sec), cosecant (cosec) and cotangent (cot) — are the reciprocals of cosine, sine and tangent respectively. They have their own identities and graphs, and are tested at A-Level (9MA0).

---

Definitions

Function	Definition	Undefined when
sec(theta)	1/cos(theta)	cos(theta) = 0, i.e. theta = pi/2 + npi
cosec(theta)	1/sin(theta)	sin(theta) = 0, i.e. theta = npi
cot(theta)	1/tan(theta) = cos(theta)/sin(theta)	sin(theta) = 0, i.e. theta = npi

---

Key Identities

Identity 1

1 + tan²(theta) = sec²(theta)

Derived from sin² + cos² = 1 by dividing by cos²:

sin²/cos² + cos²/cos² = 1/cos²

tan² + 1 = sec²

Identity 2

1 + cot²(theta) = cosec²(theta)

Derived from sin² + cos² = 1 by dividing by sin²:

sin²/sin² + cos²/sin² = 1/sin²

1 + cot² = cosec²

---

Graphs

y = sec(x)