Trigonometric identities, by family
44 identities, 10 families. Each row links to the identity page (what it is for), the proof (why it holds) and the derivation (how to find it). Everything here was sampled by the same engine as the verifier before it was published.
- Pythagorean identities · 3
- Reciprocal and quotient identities · 5
- Cofunction and even/odd identities · 6
- Sum and difference identities · 7
- Double-angle identities · 5
- Half-angle identities · 5
- Product-to-sum identities · 4
- Sum-to-product identities · 4
- Power-reducing identities · 4
- Triangle identities · 1
Pythagorean identities (3)
sin²θ + cos²θ = 1 and the two versions you get by dividing it.
| Identity | Name | details |
|---|---|---|
| sin²θ + cos²θ = 1 | Pythagorean identity | details |
| 1 + tan²θ = sec²θ | Pythagorean identity (tangent form) | details |
| 1 + cot²θ = csc²θ | Pythagorean identity (cotangent form) | details |
Reciprocal and quotient identities (5)
csc, sec, cot as reciprocals; tan = sin/cos and cot = cos/sin.
| Identity | Name | details |
|---|---|---|
| csc θ = 1/sin θ | Reciprocal identity (cosecant) | details |
| sec θ = 1/cos θ | Reciprocal identity (secant) | details |
| cot θ = 1/tan θ | Reciprocal identity (cotangent) | details |
| tan θ = sin θ / cos θ | Quotient identity (tangent) | details |
| cot θ = cos θ / sin θ | Quotient identity (cotangent) | details |
Cofunction and even/odd identities (6)
Complementary-angle pairs, and what happens to a negative angle.
| Identity | Name | details |
|---|---|---|
| sin(−θ) = −sin θ | Sine is odd | details |
| cos(−θ) = cos θ | Cosine is even | details |
| tan(−θ) = −tan θ | Tangent is odd | details |
| sin θ = cos(90° − θ) | Cofunction identity for sine | details |
| cos θ = sin(90° − θ) | Cofunction identity for cosine | details |
| tan θ = cot(90° − θ) | Cofunction identity for tangent | details |
Sum and difference identities (7)
sin(a ± b), cos(a ± b), tan(a ± b) expanded.
| Identity | Name | details |
|---|---|---|
| sin(A + B) = sin A cos B + cos A sin B | Sine of a sum | details |
| sin(A − B) = sin A cos B − cos A sin B | Sine of a difference | details |
| cos(A + B) = cos A cos B − sin A sin B | Cosine of a sum | details |
| cos(A − B) = cos A cos B + sin A sin B | Cosine of a difference | details |
| tan(A + B) = (tan A + tan B) / (1 − tan A tan B) | Tangent of a sum | details |
| tan(A − B) = (tan A − tan B) / (1 + tan A tan B) | Tangent of a difference | details |
| tan A + tan B = sin(A + B) / (cos A cos B) | Tangent sum as a single fraction | details |
Double-angle identities (5)
sin 2x, the three forms of cos 2x, and tan 2x.
| Identity | Name | details |
|---|---|---|
| sin 2x = 2 sin x cos x | Double-angle identity for sine | details |
| cos 2x = cos²x − sin²x | Double-angle identity for cosine (form 1) | details |
| cos 2x = 2 cos²x − 1 | Double-angle identity for cosine (form 2) | details |
| cos 2x = 1 − 2 sin²x | Double-angle identity for cosine (form 3) | details |
| tan 2x = 2 tan x / (1 − tan²x) | Double-angle identity for tangent | details |
Half-angle identities (5)
sin(x/2) and cos(x/2) — with the sign condition spelled out.
| Identity | Name | details |
|---|---|---|
| sin(x/2) = ±√((1 − cos x)/2) | Half-angle identity for sine | details |
| cos(x/2) = ±√((1 + cos x)/2) | Half-angle identity for cosine | details |
| tan(x/2) = (1 − cos x)/sin x | Half-angle identity for tangent (form 1) | details |
| tan(x/2) = sin x/(1 + cos x) | Half-angle identity for tangent (form 2) | details |
| tan(x/2) = csc x − cot x | Half-angle identity for tangent (form 3) | details |
Product-to-sum identities (4)
Products of sines and cosines rewritten as sums.
| Identity | Name | details |
|---|---|---|
| sin A cos B = ½[sin(A + B) + sin(A − B)] | Product to sum: sin A cos B | details |
| cos A sin B = ½[sin(A + B) − sin(A − B)] | Product to sum: cos A sin B | details |
| cos A cos B = ½[cos(A − B) + cos(A + B)] | Product to sum: cos A cos B | details |
| sin A sin B = ½[cos(A − B) − cos(A + B)] | Product to sum: sin A sin B | details |
Sum-to-product identities (4)
Sums and differences rewritten as a single product.
| Identity | Name | details |
|---|---|---|
| sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) | Sum to product: sin A + sin B | details |
| sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) | Sum to product: sin A − sin B | details |
| cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) | Sum to product: cos A + cos B | details |
| cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) | Sum to product: cos A − cos B | details |
Power-reducing identities (4)
sin², cos², sin³, cos⁴ rewritten with no powers above one.
| Identity | Name | details |
|---|---|---|
| sin²x = (1 − cos 2x)/2 | Power-reducing formula for sin² | details |
| cos²x = (1 + cos 2x)/2 | Power-reducing formula for cos² | details |
| sin³x = (3 sin x − sin 3x)/4 | Reduction formula for sin³ | details |
| cos³x = (3 cos x + cos 3x)/4 | Reduction formula for cos³ | details |
Triangle identities (1)
The area formula ½ab sin C and what it implies via the sine rule.
| Identity | Name | details |
|---|---|---|
| ½ a b sin C = ½ b c sin A | Triangle area in two pairings | details |
What is deliberately not here
- Inverse-trig relations and hyperbolic identities. Not in the catalogue yet, so they are not claimed anywhere on the site.
- Solve-for-x equations. This is an identity reference; equations are a different topic and a different tool.
- Special-angle tables alone. Those are on the cheat sheet, where they can be printed.
Prefer one page per family? Pythagorean, sum & difference, double-angle, half-angle, product-to-sum, sum-to-product, power-reducing, reciprocal & quotient, cofunction.