Deriving trig identities
A proof shows a formula is true; a derivation shows how anyone would have found it. These pages take the second route — usually "start from the sum formula and set the two angles equal", or "solve the double-angle formula for the square".
Pythagorean identities (3)
sin²θ + cos²θ = 1 and the two versions you get by dividing it.
| Identity | Name | derivation |
|---|---|---|
| sin²θ + cos²θ = 1 | Pythagorean identity | derivation |
| 1 + tan²θ = sec²θ | Pythagorean identity (tangent form) | derivation |
| 1 + cot²θ = csc²θ | Pythagorean identity (cotangent form) | derivation |
Reciprocal and quotient identities (5)
csc, sec, cot as reciprocals; tan = sin/cos and cot = cos/sin.
| Identity | Name | derivation |
|---|---|---|
| csc θ = 1/sin θ | Reciprocal identity (cosecant) | derivation |
| sec θ = 1/cos θ | Reciprocal identity (secant) | derivation |
| cot θ = 1/tan θ | Reciprocal identity (cotangent) | derivation |
| tan θ = sin θ / cos θ | Quotient identity (tangent) | derivation |
| cot θ = cos θ / sin θ | Quotient identity (cotangent) | derivation |
Cofunction and even/odd identities (6)
Complementary-angle pairs, and what happens to a negative angle.
| Identity | Name | derivation |
|---|---|---|
| sin(−θ) = −sin θ | Sine is odd | derivation |
| cos(−θ) = cos θ | Cosine is even | derivation |
| tan(−θ) = −tan θ | Tangent is odd | derivation |
| sin θ = cos(90° − θ) | Cofunction identity for sine | derivation |
| cos θ = sin(90° − θ) | Cofunction identity for cosine | derivation |
| tan θ = cot(90° − θ) | Cofunction identity for tangent | derivation |
Sum and difference identities (7)
sin(a ± b), cos(a ± b), tan(a ± b) expanded.
| Identity | Name | derivation |
|---|---|---|
| sin(A + B) = sin A cos B + cos A sin B | Sine of a sum | derivation |
| sin(A − B) = sin A cos B − cos A sin B | Sine of a difference | derivation |
| cos(A + B) = cos A cos B − sin A sin B | Cosine of a sum | derivation |
| cos(A − B) = cos A cos B + sin A sin B | Cosine of a difference | derivation |
| tan(A + B) = (tan A + tan B) / (1 − tan A tan B) | Tangent of a sum | derivation |
| tan(A − B) = (tan A − tan B) / (1 + tan A tan B) | Tangent of a difference | derivation |
| tan A + tan B = sin(A + B) / (cos A cos B) | Tangent sum as a single fraction | derivation |
Double-angle identities (5)
sin 2x, the three forms of cos 2x, and tan 2x.
| Identity | Name | derivation |
|---|---|---|
| sin 2x = 2 sin x cos x | Double-angle identity for sine | derivation |
| cos 2x = cos²x − sin²x | Double-angle identity for cosine (form 1) | derivation |
| cos 2x = 2 cos²x − 1 | Double-angle identity for cosine (form 2) | derivation |
| cos 2x = 1 − 2 sin²x | Double-angle identity for cosine (form 3) | derivation |
| tan 2x = 2 tan x / (1 − tan²x) | Double-angle identity for tangent | derivation |
Half-angle identities (5)
sin(x/2) and cos(x/2) — with the sign condition spelled out.
| Identity | Name | derivation |
|---|---|---|
| sin(x/2) = ±√((1 − cos x)/2) | Half-angle identity for sine | derivation |
| cos(x/2) = ±√((1 + cos x)/2) | Half-angle identity for cosine | derivation |
| tan(x/2) = (1 − cos x)/sin x | Half-angle identity for tangent (form 1) | derivation |
| tan(x/2) = sin x/(1 + cos x) | Half-angle identity for tangent (form 2) | derivation |
| tan(x/2) = csc x − cot x | Half-angle identity for tangent (form 3) | derivation |
Product-to-sum identities (4)
Products of sines and cosines rewritten as sums.
| Identity | Name | derivation |
|---|---|---|
| sin A cos B = ½[sin(A + B) + sin(A − B)] | Product to sum: sin A cos B | derivation |
| cos A sin B = ½[sin(A + B) − sin(A − B)] | Product to sum: cos A sin B | derivation |
| cos A cos B = ½[cos(A − B) + cos(A + B)] | Product to sum: cos A cos B | derivation |
| sin A sin B = ½[cos(A − B) − cos(A + B)] | Product to sum: sin A sin B | derivation |
Sum-to-product identities (4)
Sums and differences rewritten as a single product.
| Identity | Name | derivation |
|---|---|---|
| sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) | Sum to product: sin A + sin B | derivation |
| sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) | Sum to product: sin A − sin B | derivation |
| cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) | Sum to product: cos A + cos B | derivation |
| cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) | Sum to product: cos A − cos B | derivation |
Power-reducing identities (4)
sin², cos², sin³, cos⁴ rewritten with no powers above one.
| Identity | Name | derivation |
|---|---|---|
| sin²x = (1 − cos 2x)/2 | Power-reducing formula for sin² | derivation |
| cos²x = (1 + cos 2x)/2 | Power-reducing formula for cos² | derivation |
| sin³x = (3 sin x − sin 3x)/4 | Reduction formula for sin³ | derivation |
| cos³x = (3 cos x + cos 3x)/4 | Reduction formula for cos³ | derivation |
Triangle identities (1)
The area formula ½ab sin C and what it implies via the sine rule.
| Identity | Name | derivation |
|---|---|---|
| ½ a b sin C = ½ b c sin A | Triangle area in two pairings | derivation |