Trigonometric Identities Table
A complete reference table of trigonometric identities, organized by category. These formulas are essential tools in algebra, calculus, and physics — use them to simplify expressions, solve equations, and convert between trigonometric forms.
Fundamental Identities
The three Pythagorean identities and the basic reciprocal and quotient relationships.
| Identity | Formula |
|---|---|
| Pythagorean (sin/cos) | sin²θ + cos²θ = 1 |
| Pythagorean (tan/sec) | 1 + tan²θ = sec²θ |
| Pythagorean (cot/csc) | 1 + cot²θ = csc²θ |
| Reciprocal — csc | csc θ = 1 / sin θ |
| Reciprocal — sec | sec θ = 1 / cos θ |
| Reciprocal — cot | cot θ = 1 / tan θ |
| Quotient — tan | tan θ = sin θ / cos θ |
| Quotient — cot | cot θ = cos θ / sin θ |
Cofunction and Even/Odd Identities
Cofunction identities relate a function to its complement (90° − θ). Even/odd identities describe behavior under negation of the angle.
| Identity | Formula |
|---|---|
| Cofunction — sin/cos | sin θ = cos(90° − θ) |
| Cofunction — cos/sin | cos θ = sin(90° − θ) |
| Cofunction — tan/cot | tan θ = cot(90° − θ) |
| Cofunction — cot/tan | cot θ = tan(90° − θ) |
| Cofunction — sec/csc | sec θ = csc(90° − θ) |
| Cofunction — csc/sec | csc θ = sec(90° − θ) |
| Even — cos | cos(−θ) = cos θ |
| Even — sec | sec(−θ) = sec θ |
| Odd — sin | sin(−θ) = −sin θ |
| Odd — tan | tan(−θ) = −tan θ |
| Odd — csc | csc(−θ) = −csc θ |
| Odd — cot | cot(−θ) = −cot θ |
Sum and Difference Formulas
Expand or simplify expressions involving sums or differences of two angles.
| Identity | Formula |
|---|---|
| sin(A + B) | sin A cos B + cos A sin B |
| sin(A − B) | sin A cos B − cos A sin B |
| cos(A + B) | cos A cos B − sin A sin B |
| cos(A − B) | cos A cos B + sin A sin B |
| tan(A + B) | (tan A + tan B) / (1 − tan A tan B) |
| tan(A − B) | (tan A − tan B) / (1 + tan A tan B) |
Double Angle Formulas
Express functions of 2A in terms of functions of A. cos(2A) has three equivalent forms.
| Identity | Formula |
|---|---|
| sin(2A) | 2 sin A cos A |
| cos(2A) — form 1 | cos²A − sin²A |
| cos(2A) — form 2 | 2 cos²A − 1 |
| cos(2A) — form 3 | 1 − 2 sin²A |
| tan(2A) | 2 tan A / (1 − tan²A) |
Half Angle Formulas
Express functions of A/2 in terms of functions of A. The ± sign depends on the quadrant of A/2.
| Identity | Formula |
|---|---|
| sin(A/2) | ±√((1 − cos A) / 2) |
| cos(A/2) | ±√((1 + cos A) / 2) |
| tan(A/2) — form 1 | ±√((1 − cos A) / (1 + cos A)) |
| tan(A/2) — form 2 | sin A / (1 + cos A) |
| tan(A/2) — form 3 | (1 − cos A) / sin A |
Power-Reducing Formulas
Reduce the power of squared trig functions. Derived from the double angle formulas for cos(2A).
| Identity | Formula |
|---|---|
| sin²A | (1 − cos 2A) / 2 |
| cos²A | (1 + cos 2A) / 2 |
| tan²A | (1 − cos 2A) / (1 + cos 2A) |
Product-to-Sum and Sum-to-Product
Convert between products and sums of trigonometric functions.
| Identity | Formula |
|---|---|
| sin A cos B | ½ [sin(A + B) + sin(A − B)] |
| cos A cos B | ½ [cos(A − B) + cos(A + B)] |
| sin A sin B | ½ [cos(A − B) − cos(A + B)] |
| sin A + sin B | 2 sin((A+B)/2) cos((A−B)/2) |
| sin A − sin B | 2 cos((A+B)/2) sin((A−B)/2) |
| cos A + cos B | 2 cos((A+B)/2) cos((A−B)/2) |
| cos A − cos B | −2 sin((A+B)/2) sin((A−B)/2) |