Half-angle identities · trigonometric identity
tan(x/2) = csc x − cot x — Half-angle identity for tangent (form 3)
When to use it
The compact textbook form, and the one that appears when integrating csc and cot. It is the same identity wearing reciprocal names.
Why it is true
Put (1 − cos x)/sin x over the common denominator sin x: 1/sin x − cos x/sin x, which is csc x − cot x.
The full line-by-line version is on the proof page for half-angle identity for tangent (form 3); the “how would I find this myself” version is in the derivation.
Where it comes from
Read the names off the pieces. If you can derive form 1, this one is only naming: 1/sin is csc, cos/sin is cot.
- 1/sin x is csc xReciprocal. · Reciprocal identity (cosecant)
- cos x/sin x is cot xQuotient. · Quotient identity (cotangent)
- Difference of the twoConclusion.
Worked examples
- Show that the three half-angle tangent forms agree at x = 60°All three give 1/√3 = tan 30°
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 |
|---|---|---|---|
| 7.5° | 0.065543 | 0.065543 | yes |
| 15° | 0.131652 | 0.131652 | yes |
| 18° | 0.158384 | 0.158384 | yes |
| 22.5° | 0.198912 | 0.198912 | yes |
| 30° | 0.267949 | 0.267949 | yes |
| 37° | 0.334595 | 0.334595 | yes |
| 45° | 0.414214 | 0.414214 | yes |
| 53° | 0.498582 | 0.498582 | yes |
| 60° | 0.57735 | 0.57735 | yes |
| 67.5° | 0.668179 | 0.668179 | yes |
| 75° | 0.767327 | 0.767327 | yes |
| 90° | 1 | 1 | yes |
| 120° | 1.732051 | 1.732051 | yes |
| 135° | 2.414214 | 2.414214 | yes |
| 150° | 3.732051 | 3.732051 | yes |
| 210° | -3.732051 | -3.732051 | yes |
| 240° | -1.732051 | -1.732051 | yes |
| 270° | -1 | -1 | yes |
| 300° | -0.57735 | -0.57735 | yes |
| 330° | -0.267949 | -0.267949 | yes |
The mistake students make
Missing that this form hides the quotient structure; if you need to combine it with other fractions, expand back to (1 − cos x)/sin x first.
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 half-angle identities.
Related identities
- tan(x/2) = (1 − cos x)/sin xHalf-angle identity for tangent (form 1)detailsproof
- tan(x/2) = sin x/(1 + cos x)Half-angle identity for tangent (form 2)detailsproof
- csc θ = 1/sin θReciprocal identity (cosecant)detailsproof
- cot θ = cos θ / sin θQuotient identity (cotangent)detailsproof
Category hub: Half-angle identities · all identities: /identities