Double-angle identities · derivation ·proof
Where tan 2x = 2 tan x / (1 − tan²x) comes from
Or divide the other two. If the tangent formula is not memorised: tan 2x = sin 2x / cos 2x, then substitute the sine and cosine double-angle forms and divide top and bottom by cos²x.
tan 2x = 2 tan x / (1 − tan²x)
Building it step by step
- tan 2x = sin 2x/cos 2xQuotient identity. · Quotient identity (tangent)
- = 2 sin x cos x/(cos²x − sin²x)Both double-angle forms. · Double-angle identity for sine, Double-angle identity for cosine (form 1)
- divide by cos²xTurns each piece into a tangent.
- = 2 tan x/(1 − tan²x)Result.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- tan θ = sin θ / cos θ — Quotient identity (tangent)
- sin 2x = 2 sin x cos x — Double-angle identity for sine
- cos 2x = cos²x − sin²x — Double-angle identity for cosine (form 1)
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from tan θ = sin θ / cos θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the double-angle identity for tangent page.