Worked example

Show that the three half-angle tangent forms agree at x = 60°

Steps

1. tan 30° = 1/√3
   Target — the left side at x = 60°.
2. (1 − cos 60°)/sin 60° = (1 − 1/2)/(√3/2) = 1/√3
   Form 1. · Half-angle identity for tangent (form 1)
3. sin 60°/(1 + cos 60°) = (√3/2)/(3/2) = 1/√3
   Form 2. · Half-angle identity for tangent (form 2)
4. csc 60° − cot 60° = 2/√3 − 1/√3 = 1/√3
   Form 3 — three roads, one value. · Half-angle identity for tangent (form 3)

Identities used

• Half-angle identity for tangent (form 1) — 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.
• Half-angle identity for tangent (form 2) — The companion form; prefer it when the numerator sin x is already in your expression, and the one that leads directly to the tangent half-angle substitution used in integration.
• Half-angle identity for tangent (form 3) — The compact textbook form, and the one that appears when integrating csc and cot. It is the same identity wearing reciprocal names.

Check it yourself

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.

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