Complete Trig Identities

Pythagorean, Angle, and Reciprocal Identities Sum, Difference, Double, and Half Angle Identities

Pythagorean Identity (sin & cos)

sin²θ + cos²θ = 1

Pythagorean Identity (tan & sec)

1 + tan²θ = sec²θ

Pythagorean Identity (cot & csc)

1 + cot²θ = csc²θ

Reciprocal Identity (csc)

cscθ = 1 / sinθ

Reciprocal Identity (sec)

secθ = 1 / cosθ

Reciprocal Identity (cot)

cotθ = 1 / tanθ

Quotient Identity (tan)

tanθ = sinθ / cosθ

Quotient Identity (cot)

cotθ = cosθ / sinθ

Co-Function Identity (sin & cos)

sin(90° − θ) = cosθ

Co-Function Identity (cos & sin)

cos(90° − θ) = sinθ

Co-Function Identity (tan & cot)

tan(90° − θ) = cotθ

Co-Function Identity (cot & tan)

cot(90° − θ) = tanθ

Co-Function Identity (sec & csc)

sec(90° − θ) = cscθ

Co-Function Identity (csc & sec)

csc(90° − θ) = secθ

Even/Odd Identity (sin)

sin(−θ) = −sinθ

Even/Odd Identity (cos)

cos(−θ) = cosθ

Even/Odd Identity (tan)

tan(−θ) = −tanθ

Even/Odd Identity (csc)

csc(−θ) = −cscθ

Even/Odd Identity (sec)

sec(−θ) = secθ

Even/Odd Identity (cot)

cot(−θ) = −cotθ

Sum Formula (sin(A + B))

sin(A + B) = sinA cosB + cosA sinB

Sum Formula (cos(A + B))

cos(A + B) = cosA cosB − sinA sinB

Sum Formula (tan(A + B))

tan(A + B) = (tanA + tanB) / (1 − tanA tanB)

Difference Formula (sin(A − B))

sin(A − B) = sinA cosB − cosA sinB

Difference Formula (cos(A − B))

cos(A − B) = cosA cosB + sinA sinB

Difference Formula (tan(A − B))

tan(A − B) = (tanA − tanB) / (1 + tanA tanB)

Double-Angle (sin(2θ))

sin(2θ) = 2 sinθ cosθ

Double-Angle (cos(2θ) v1)

cos(2θ) = cos²θ − sin²θ

Double-Angle (cos(2θ) v2)

cos(2θ) = 1 − 2sin²θ

Double-Angle (cos(2θ) v3)

cos(2θ) = 2cos²θ − 1

Double-Angle (tan(2θ))

tan(2θ) = (2 tanθ) / (1 − tan²θ)

Half-Angle (sin(θ/2))

sin(θ/2) = ±√[(1 − cosθ) / 2]

Half-Angle (cos(θ/2))

cos(θ/2) = ±√[(1 + cosθ) / 2]

Half-Angle (tan(θ/2) root)

tan(θ/2) = ±√[(1 − cosθ) / (1 + cosθ)]

Half-Angle (tan(θ/2) sin/cos)

tan(θ/2) = sinθ / (1 + cosθ)

Half-Angle (tan(θ/2) alt)

tan(θ/2) = (1 − cosθ) / sinθ