Cofunction and even/odd identities · trigonometric identity
cos(−θ) = cos θ — Cosine is even
When to use it
Drop the minus sign inside a cosine outright. Useful when simplifying cos(−x) + sin(−x), where the cosine keeps its sign and the sine does not.
Why it is true
Reflecting a unit-circle point across the x-axis does not change its x-coordinate, and that x-coordinate is the cosine.
The full line-by-line version is on the proof page for cosine is even; the “how would I find this myself” version is in the derivation.
Where it comes from
Numbers first. cos 30° = cos(−30°) = √3/2: the wave is symmetric about the y-axis.
- cos 30° = √3/2Special angle.
- cos(−30°) = √3/2Mirror point, same x-coordinate.
- cos(−θ) = cos θSymmetry of the graph about the y-axis.
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° | 1 | 1 | yes |
| 7.5° | 0.991445 | 0.991445 | yes |
| 15° | 0.965926 | 0.965926 | yes |
| 18° | 0.951057 | 0.951057 | yes |
| 22.5° | 0.92388 | 0.92388 | yes |
| 30° | 0.866025 | 0.866025 | yes |
| 37° | 0.798636 | 0.798636 | yes |
| 45° | 0.707107 | 0.707107 | yes |
| 53° | 0.601815 | 0.601815 | yes |
| 60° | 0.5 | 0.5 | yes |
| 67.5° | 0.382683 | 0.382683 | yes |
| 75° | 0.258819 | 0.258819 | yes |
| 90° | 0 | 0 | yes |
| 120° | -0.5 | -0.5 | yes |
| 135° | -0.707107 | -0.707107 | yes |
| 150° | -0.866025 | -0.866025 | yes |
| 180° | -1 | -1 | yes |
| 210° | -0.866025 | -0.866025 | yes |
| 240° | -0.5 | -0.5 | yes |
| 270° | -0 | -0 | yes |
| 300° | 0.5 | 0.5 | yes |
| 330° | 0.866025 | 0.866025 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Expecting every function to behave like sine. Secant is even too (it is 1/cos); tangent, cotangent and cosecant are odd.
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
- tan(−θ) = −tan θTangent is odddetailsproof
- sec θ = 1/cos θReciprocal identity (secant)detailsproof
Category hub: Cofunction and even/odd identities · all identities: /identities