Cofunction and even/odd identities · trigonometric identity
tan(−θ) = −tan θ — Tangent is odd
When to use it
Pull the minus out of tan(−θ). Also the quickest way to see why tan is odd: sin flips, cos does not, so the ratio flips.
Why it is true
tan θ = sin θ/cos θ. Under θ → −θ the numerator changes sign and the denominator does not, so the quotient changes sign.
The full line-by-line version is on the proof page for tangent is odd; the “how would I find this myself” version is in the derivation.
Where it comes from
Odd divided by even. Rule of thumb worth remembering: a ratio whose top flips sign flips sign too.
- top: odd, bottom: evensin is odd, cos is even.
- ratio: odd−a/b = −(a/b).
- tan(−θ) = −tan θApply to θ.
Worked examples
- Simplify sin(−x)cos(−x) + tan(−x)−sin x cos x − tan x
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° | 0 | 0 | yes |
| 7.5° | -0.131652 | -0.131652 | yes |
| 15° | -0.267949 | -0.267949 | yes |
| 18° | -0.32492 | -0.32492 | yes |
| 22.5° | -0.414214 | -0.414214 | yes |
| 30° | -0.57735 | -0.57735 | yes |
| 37° | -0.753554 | -0.753554 | yes |
| 45° | -1 | -1 | yes |
| 53° | -1.327045 | -1.327045 | yes |
| 60° | -1.732051 | -1.732051 | yes |
| 67.5° | -2.414214 | -2.414214 | yes |
| 75° | -3.732051 | -3.732051 | yes |
| 120° | 1.732051 | 1.732051 | yes |
| 135° | 1 | 1 | yes |
| 150° | 0.57735 | 0.57735 | yes |
| 180° | 0 | 0 | yes |
| 210° | -0.57735 | -0.57735 | yes |
| 240° | -1.732051 | -1.732051 | yes |
| 300° | 1.732051 | 1.732051 | yes |
| 330° | 0.57735 | 0.57735 | yes |
| 360° | 0 | 0 | yes |
The mistake students make
Forgetting the excluded angles: tan(−θ) is undefined exactly where tan θ is undefined (cos θ = 0), and the sign rule says nothing there.
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 cofunction and even/odd identities.
Related identities
- sin(−θ) = −sin θSine is odddetailsproof
- cos(−θ) = cos θCosine is evendetailsproof
- cot θ = 1/tan θReciprocal identity (cotangent)detailsproof
Category hub: Cofunction and even/odd identities · all identities: /identities