Pythagorean identities · trigonometric identity
1 + tan²θ = sec²θ — Pythagorean identity (tangent form)
When to use it
You see 1 + tan²θ — most often under a square root, √(1 + tan²θ), or as the product sec²θ appearing in a derivative. Replace the whole sum with sec²θ, or a lone sec²θ with 1 + tan²θ when the tangent form is what you need.
Why it is true
It is sin²θ + cos²θ = 1 divided through by cos²θ. The 1 comes from cos²θ/cos²θ, the tan²θ from sin²θ/cos²θ, and the right side is 1/cos²θ = sec²θ.
The full line-by-line version is on the proof page for pythagorean identity (tangent form); the “how would I find this myself” version is in the derivation.
Where it comes from
Same division, aimed forward. If you have forgotten the formula, re-derive it in ten seconds: you want a relation involving tan, so force tan = sin/cos to appear by dividing by cos²θ.
- Want tan²θ? Divide by cos²θ.tan²θ = sin²θ/cos²θ, so the division creates exactly the term you need.
- 1/cos²θ is sec²θReciprocal identity. · Reciprocal identity (secant)
- 1 + tan²θ = sec²θCollect the three pieces.
Worked examples
- Verify (1 + tan²θ)cos θ = sec θLeft side reduces to 1/cos θ = sec θ, so the equation is an identity.
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° | 1.017332 | 1.017332 | yes |
| 15° | 1.071797 | 1.071797 | yes |
| 18° | 1.105573 | 1.105573 | yes |
| 22.5° | 1.171573 | 1.171573 | yes |
| 30° | 1.333333 | 1.333333 | yes |
| 37° | 1.567844 | 1.567844 | yes |
| 45° | 2 | 2 | yes |
| 53° | 2.761048 | 2.761048 | yes |
| 60° | 4 | 4 | yes |
| 67.5° | 6.828427 | 6.828427 | yes |
| 75° | 14.928203 | 14.928203 | yes |
| 120° | 4 | 4 | yes |
| 135° | 2 | 2 | yes |
| 150° | 1.333333 | 1.333333 | yes |
| 180° | 1 | 1 | yes |
| 210° | 1.333333 | 1.333333 | yes |
| 240° | 4 | 4 | yes |
| 300° | 4 | 4 | yes |
| 330° | 1.333333 | 1.333333 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Believing the identity is true for every angle. Both sides blow up when cos θ = 0, so θ = 90° is excluded — the two sides are undefined together, not equal.
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 pythagorean identities.
Related identities
- sin²θ + cos²θ = 1Pythagorean identitydetailsproof
- 1 + cot²θ = csc²θPythagorean identity (cotangent form)detailsproof
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
- sec θ = 1/cos θReciprocal identity (secant)detailsproof
Category hub: Pythagorean identities · all identities: /identities