Reciprocal and quotient identities · trigonometric identity
sec θ = 1/cos θ — Reciprocal identity (secant)
When to use it
Convert sec θ to 1/cos θ whenever it shares an expression with tan θ, since tan θ is already sin/cos and a common denominator then appears.
Why it is true
Secant is the reciprocal of cosine by definition: cos θ = adjacent/hypotenuse, so hypotenuse/adjacent = sec θ.
The full line-by-line version is on the proof page for reciprocal identity (secant); the “how would I find this myself” version is in the derivation.
Where it comes from
Flip and name. Secant is 'the reciprocal of cosine' wearing a different name; the derivation is the flip.
- cos θ = adj/hypRight-triangle ratio.
- 1/cos θ = hyp/adjAlgebra.
- hyp/adj is sec θNaming.
Worked examples
- Verify or refute: sin²θ · sec²θ − 1 = tan²θsin²θ · sec²θ − 1 = tan²θ, an identity (true for every θ where defined).
- Verify (1 + tan²θ)cos θ = sec θLeft side reduces to 1/cos θ = sec θ, so the equation is an identity.
- Verify tan θ + cot θ = sec θ csc θConvert to sin and cos, take the common denominator, and the Pythagorean identity closes it.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 1 | 1 | yes |
| 7.5° | 1.008629 | 1.008629 | yes |
| 15° | 1.035276 | 1.035276 | yes |
| 18° | 1.051462 | 1.051462 | yes |
| 22.5° | 1.082392 | 1.082392 | yes |
| 30° | 1.154701 | 1.154701 | yes |
| 37° | 1.252136 | 1.252136 | yes |
| 45° | 1.414214 | 1.414214 | yes |
| 53° | 1.66164 | 1.66164 | yes |
| 60° | 2 | 2 | yes |
| 67.5° | 2.613126 | 2.613126 | yes |
| 75° | 3.863703 | 3.863703 | yes |
| 120° | -2 | -2 | yes |
| 135° | -1.414214 | -1.414214 | yes |
| 150° | -1.154701 | -1.154701 | yes |
| 180° | -1 | -1 | yes |
| 210° | -1.154701 | -1.154701 | yes |
| 240° | -2 | -2 | yes |
| 300° | 2 | 2 | yes |
| 330° | 1.154701 | 1.154701 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Treating sec θ as sec·θ, or confusing it with cos⁻¹, which usually means the inverse function (arccos), not the reciprocal. Here 1/cos θ is the reciprocal; arccos θ is the angle.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for reciprocal and quotient identities.
Related identities
- csc θ = 1/sin θReciprocal identity (cosecant)detailsproof
- cot θ = 1/tan θReciprocal identity (cotangent)detailsproof
- 1 + tan²θ = sec²θPythagorean identity (tangent form)detailsproof
Category hub: Reciprocal and quotient identities · all identities: /identities