Complete List of Trigonometric Identities

This is the complete guide to all Trigonometric identities, memorize here, remember everything forever.

Fundamental Pythagorean Identity

sin²(x) + cos²(x) = 1

Pythagorean Identity with tan and sec

1 + tan²(x) = sec²(x)

Pythagorean Identity with cot and csc

1 + cot²(x) = csc²(x)

sin(θ) in terms of its reciprocal

sin(θ) = 1/csc(θ)

cos(θ) in terms of its reciprocal

cos(θ) = 1/sec(θ)

tan(θ) in terms of its reciprocal

tan(θ) = 1/cot(θ)

csc(θ) in terms of its reciprocal

csc(θ) = 1/sin(θ)

sec(θ) in terms of its reciprocal

sec(θ) = 1/cos(θ)

cot(θ) in terms of its reciprocal

cot(θ) = 1/tan(θ)

Is sin(x) even or odd?

sin(-x) = -sin(x) (odd)

Is cos(x) even or odd?

cos(-x) = cos(x) (even)

Is tan(x) even or odd?

tan(-x) = -tan(x) (odd)

Is csc(x) even or odd?

csc(-x) = -csc(x) (odd)

Is sec(x) even or odd?

sec(-x) = sec(x) (even)

Is cot(x) even or odd?

cot(-x) = -cot(x) (odd)

sin(π/2 - x)

sin(π/2 - x) = cos(x)

cos(π/2 - x)

cos(π/2 - x) = sin(x)

tan(π/2 - x)

tan(π/2 - x) = cot(x)

cot(π/2 - x)

cot(π/2 - x) = tan(x)

sec(π/2 - x)

sec(π/2 - x) = csc(x)

csc(π/2 - x)

csc(π/2 - x) = sec(x)

sin(a + b)

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

sin(a - b)

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

cos(a + b)

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

cos(a - b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

tan(a + b)

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

tan(a - b)

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

sin(2x)

sin(2x) = 2sin(x)cos(x)

cos(2x) (primary form)

cos(2x) = cos²(x) - sin²(x)

Alternative form for cos(2x)

cos(2x) = 2cos²(x) - 1 = 1 - 2sin²(x)

tan(2x)

tan(2x) = (2tan(x)) / (1 - tan²(x))

sin(3x)

sin(3x) = 3sin(x) - 4sin³(x)

cos(3x)

cos(3x) = 4cos³(x) - 3cos(x)

tan(3x)

tan(3x) = (3tan(x) - tan³(x)) / (1 - 3tan²(x))

sin(x/2) in terms of cos(x)

sin(x/2) = ±√((1 - cos(x)) / 2)

cos(x/2) in terms of cos(x)

cos(x/2) = ±√((1 + cos(x)) / 2)

tan(x/2) (common form)

tan(x/2) = sin(x) / (1 + cos(x)) or (1 - cos(x)) / sin(x)

Product-to-Sum: sin(A)sin(B)

sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]

Product-to-Sum: cos(A)cos(B)

cos(A)cos(B) = 1/2[cos(A - B) + cos(A + B)]

Product-to-Sum: sin(A)cos(B)

sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]

Sum-to-Product: sin(A) + sin(B)

sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)

Sum-to-Product: sin(A) - sin(B)

sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)

Sum-to-Product: cos(A) + cos(B)

cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)

Sum-to-Product: cos(A) - cos(B)

cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)

Power-Reducing: sin²(x)

sin²(x) = (1 - cos(2x))/2

Power-Reducing: cos²(x)

cos²(x) = (1 + cos(2x))/2

Power-Reducing: tan²(x)

tan²(x) = (1 - cos(2x))/(1 + cos(2x))

De Moivre’s Theorem

(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ)

Euler’s Formula

e^(iθ) = cos(θ) + isin(θ)

Combine asin(x) + bcos(x)

asin(x) + bcos(x) = Rsin(x + φ) where R = √(a² + b²), tan(φ) = b/a.

arcsin(x ± y)

arcsin(x ± y) = arcsin(x) ± arcsin(y) (Valid where x ± y is within the range of arcsin function)

arccos(x ± y)

arccos(x ± y) = arccos(x) ± arccos(y) (Valid where x ± y is within the range of arccos function)

Periodic Identity of sin(x)

sin(x) has a period of 2π, meaning sin(x + 2π) = sin(x) for all x.

Periodic Identity of cos(x)

cos(x) has a period of 2π, meaning cos(x + 2π) = cos(x) for all x.

Periodic Identity of csc(x)

csc(x) has a period of 2π, meaning csc(x + 2π) = csc(x) for all x.

Periodic Identity of sec(x)

sec(x) has a period of 2π, meaning sec(x + 2π) = sec(x) for all x.

Periodic Identity of tan(x)

tan(x) has a period of π, meaning tan(x + π) = tan(x) for all x.

Periodic Identity of cot(x)

cot(x) has a period of π, meaning cot(x + π) = cot(x) for all x.

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) for triangle with sides a, b, c opposite angles A, B, C respectively.

Law of Cosines (version 1)

c² = a² + b² - 2ab * cos(C)

Law of Cosines (version 2)

a² = b² + c² - 2bc * cos(A)

Law of Cosines (version 3)

b² = a² + c² - 2ac * cos(B)

Law of Tangents (version 1)

(a - b) / (a + b) = tan((A - B) / 2) / tan((A + B) / 2)

Law of Tangents (version 2)

(b - c) / (b + c) = tan((B - C) / 2) / tan((B + C) / 2)

Law of Tangents (version 3)

(c - a) / (c + a) = tan((C - A) / 2) / tan((C + A) / 2)

Cofunction Identity for Sin and Cos

sin(π/2 - x) = cos(x)

Cofunction Identity for Cos and Sin

cos(π/2 - x) = sin(x)

Cofunction Identity for Tan and Cot

tan(π/2 - x) = cot(x)

Cofunction Identity for Cot and Tan

cot(π/2 - x) = tan(x)

Cofunction Identity for Sec and Csc

sec(π/2 - x) = csc(x)

Cofunction Identity for Csc and Sec

csc(π/2 - x) = sec(x)

Mollweide's Formula

For a triangle with angles A, B, and C, and opposite sides a, b, and c, Mollweide's formula states: asin(A) + bsin(B) = c*sin(C).

Double Angle Identity for Sine

sin(2x) = 2sin(x)cos(x)

Double Angle Identity for Cosine

cos(2x) = cos²(x) - sin²(x) = 2cos²(x) - 1 = 1 - 2sin²(x)

Half Angle Identity for Sine

sin(x/2) = ±√((1 - cos(x)) / 2)

Half Angle Identity for Cosine

cos(x/2) = ±√((1 + cos(x)) / 2)

Half Angle Identity for Tangent

tan(x/2) = sin(x) / (1 + cos(x)) = (1 - cos(x)) / sin(x)