Reference & tools · 10 identity families
Which trig identity should I use?
A trigonometric identity is an equation that is true for every angle — not something to solve, something to swap. The one everything else grows from is sin²θ + cos²θ = 1 : the point at angle θ on a unit circle sits at (cos θ, sin θ), so Pythagoras forces the squares to add to 1. Divide that by cos²θ or sin²θ and you get the two other Pythagorean forms; set the two angles in sin(A + B) equal and you get the double-angle family. Students collect these as a list to memorise; they are a short chain to recognise, and recognition is the skill being tested.
Is this claim actually true?
Paste both sides; the tool samples angles and, if the claim is false, stops at the first angle where it fails and shows you the numbers.
This page's own run of that engine: sin²θ + cos²θ = 1 agreed at 100 of 100 sampled angles (worst relative difference 0.0e+0).
The families
Sum and difference identities
sin(a ± b), cos(a ± b), tan(a ± b) expanded.
7 identitys · proof and derivation for each
Cofunction and even/odd identities
Complementary-angle pairs, and what happens to a negative angle.
6 identitys · proof and derivation for each
Reciprocal and quotient identities
csc, sec, cot as reciprocals; tan = sin/cos and cot = cos/sin.
5 identitys · proof and derivation for each
Double-angle identities
sin 2x, the three forms of cos 2x, and tan 2x.
5 identitys · proof and derivation for each
Half-angle identities
sin(x/2) and cos(x/2) — with the sign condition spelled out.
5 identitys · proof and derivation for each
Product-to-sum identities
Products of sines and cosines rewritten as sums.
4 identitys · proof and derivation for each
Sum-to-product identities
Sums and differences rewritten as a single product.
4 identitys · proof and derivation for each
Power-reducing identities
sin², cos², sin³, cos⁴ rewritten with no powers above one.
4 identitys · proof and derivation for each
Pythagorean identities
sin²θ + cos²θ = 1 and the two versions you get by dividing it.
3 identitys · proof and derivation for each
Triangle identities
The area formula ½ab sin C and what it implies via the sine rule.
1 identity · proof and derivation for each
Questions that have no good page anywhere
Most searches in this subject are one narrow question — “why does cos 2x have three forms?”, “when is the half-angle negative?”. Those get their own addresses here.
- Why is sin²θ + cos²θ = 1 true? — sin²θ + cos²θ = 1
- Where does sin 2x = 2 sin x cos x come from? — sin 2x = 2 sin x cos x
- How do I turn sin 3x cos x into something I can integrate? — sin A cos B = ½[sin(A + B) + sin(A − B)]
- How do I choose the sign in a half-angle problem? — worked with 180° < x < 270°
- All 28 worked examples
Other tools
- Product to sum and sum to product converters, with the check that the conversion is valid
- Degrees ↔ radians — with exact multiples of π
- Printable cheat sheet — one page, your choice of families
- Practice problems — 0 with hints and answers
- Complete identity list · proofs · derivations
44 identities, 28 worked examples. Each one is evaluated at up to 800 random angles by the same engine the tools run before the site will build at all — the method and its limits are on the accuracy policy page. Data version 2026.09.26-1, sources checked 2026-09-26.