Reciprocal and quotient identities · proof ·what it is and when to use it
Proof: sec θ = 1/cos θ
Reciprocal identity (secant) — proved by definition. Every line below says which rule it uses, so nothing has to be taken on faith.
sec θ = 1/cos θ
The proof
Same argument as for cosecant, with the other leg of the triangle.
- cos θ = adj/hypCAH.
- hyp/adj = 1/cos θReciprocal of both sides.
- sec θ = hyp/adjDefinition of secant.
Where the proof stops applying
Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| 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 |
Related
- How would I find sec θ = 1/cos θ myself? — the derivation, which is a different question from the proof.
- Reciprocal identity (secant): when to use it — the practical side.
- All reciprocal and quotient identities · proof index