Half-angle identities · trigonometric identity
tan(x/2) = (1 − cos x)/sin x — Half-angle identity for tangent (form 1)
When to use it
The root-free half-angle tangent. Because there is no square root, there is no sign question — this form is the one to use when you want a definite expression.
Why it is true
Write tan(x/2) = sin(x/2)/cos(x/2), then multiply top and bottom by cos(x/2) and use the double-angle formulas: the numerator becomes sin x and the denominator 1 + cos x — the other form. Multiply instead by sin(x/2) to get this one.
The full line-by-line version is on the proof page for half-angle identity for tangent (form 1); the “how would I find this myself” version is in the derivation.
Where it comes from
Rationalise to switch forms. The two tangent half-angle forms are the same fraction with the conjugate multiplied in — you never need to memorise both.
- Start with sin x/(1 + cos x)One form.
- Multiply by (1 − cos x)/(1 − cos x)Rationalising factor.
- Bottom: 1 − cos²x = sin²xPythagorean identity. · Pythagorean identity
- Cancel one sin x → (1 − cos x)/sin xThe other form. · Half-angle identity for tangent (form 2)
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
Undefined points: sin x = 0 means x = 0° or 180°, and at those the half-angle tangent is 0 or undefined. The formula is not claiming values 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 half-angle identities.
Related identities
- tan(x/2) = sin x/(1 + cos x)Half-angle identity for tangent (form 2)detailsproof
- tan(x/2) = csc x − cot xHalf-angle identity for tangent (form 3)detailsproof
- sin 2x = 2 sin x cos xDouble-angle identity for sinedetailsproof
- sin²θ + cos²θ = 1Pythagorean identitydetailsproof
Category hub: Half-angle identities · all identities: /identities