Basic Definitions

This is a list of commonly used trig identities. Everything a beginning to intermediate student would need is here. While an exhaustive list might be helpful, all other identities can be derived from these. Even most of the following can be derived from each other.

Quotient Identities:

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)

Reciprocal Functions:

sin(x) = 1/csc(x)

cos(x) = 1/sec(x)

tan(x) = 1/cot(x)

csc(x) = 1/sin(x)

sec(x) = 1/cos(x)

cot(x) = 1/tan(x)

Pythagorean Identities:

sin²(x) + cos2(x) = 1

1+tan²(x) = sec²(x)

cot²(x) + 1 = csc²(x)

Inverse Functions:

Remember that trig inverses are only partial inverse functions, and don’t cover the entire number line.

sin^-1(x) = arcsin(x)

cos^-1(x) = arccos(x)

tan^-1(x) = arctan(x)

csc^-1(x) = arccsc(x)

sec^-1(x) = arcsec(x)

cot^-1(x) = arccot(x)

Co-Functions, Even-Odd and Sum-Difference Formulas

Co-Function Identities:

(here, p = pi radians = 3.14159 radians = 180 degrees)

sin(p/2 – x) = cos(x)

cos(p/2 – x) = sin(x)

tan(p/2 – x) = cot(x)

csc(p/2 – x) = sec(x)

sec(p/2 – x) = csc(x)

cot(p/2 – x) = tan(x)

Even-Odd Identities

sin(-x) = -sin(x)

cos(-x) = cos(x)

tan(-x) = -tan(x)

csc(-x) = -csc(x)

sec(-x) = sec(x)

cot(-x) = -cot(x)

Sum-Difference Formulas

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

sin(x-y) = sin(x)cos(y) – cos(x)sin(y)

cos(x+y) = cos(x)cos(y) – sin(x)sin(y)

cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

tan(x+y) = [tan(x) + tan(y)] / [1 – tan(x)tan(y)]

tan(x-y) = [tan(x) – tan(y)] / [1 + tan(x)tan(y)]

Sum-Difference Formulas for Inverses:

arcsin(x) + arcsin(y) = arcsin[x*sqrt(1-y²) + y*sqrt(1-x²)]

arcsin(x) – arcsin(y) = arcsin[x*sqrt(1-y²) – y*sqrt(1-x²)]

arccos(x) + arccos(y) = arccos(xy – sqrt[(1-x²)*(1-y²)]

arccos(x) – arccos(y) = arccos(xy + sqrt[(1-x²)*(1-y²)]

arctan(x) + arctan(y) = arctan([x+y] / [1-xy])

arctan(x) – arctan(y) = arctan([x-y] / [1+xy])

Tangent of an Average:

tan([x+y]/2) = [sinx+siny]/[cosx+cosy]

tan([x+y]/2) = – ([cosx-cosy] / [sin(x)-sin(y)])

Double Angle, Power-Reduction and Half Angle Formulas

Double Angle Formulas:

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

cos(2x) = cos²(x) -sin²(x)

cos(2x) = 2cos²(x)-1

cos(2x) = 1-2sin²(x)

tan(2x) = [2tan(x)] / [1-tan²(x)]

Power-Reduction Formulas:

sin²(x) = (1/2)*[1-cos(2x)]

cos²(x) = (1/2)*[1+cos(2x)]

tan²(x) = [1-cos(2x)] / [1+cos(2x)]

Half Angle Formulas:

sin(x/2) = ~+mn~sqrt [(1/2)*(1-cos(x))]

cos(x/2) = ~+mn~sqrt [(1/2)*(1+cos(x))]

tan(x/2) = ~+mn~sqrt [(1-cos(x)) / (1+cos(x))]

tan(x/2) = sin(x) / [1+cos(x)]

tan(x/2) = [1-cos(x)]/sin(x)

Sum-to-Product and Product-to-Sum Formulas

Sum-to-Product Formulas:

sin(x) + sin(y) = 2*sin([x+y]/2)*cos([x-y]/2)

sin(x) – sin(y) = 2*cos([x+y]/2)*sin([x-y]/2)

cos(x) + cos(y) = 2*cos([x+y]/2)*cos([x-y]/2)

cos(x) + cos(y) = 2*sin([x+y]/2)*sin([x-y]/2)

Product-to-Sum Formulas:

sin(x)sin(y) = (1/2)*[cos(x-y) – cos(x+y)]

cos(x)cos(y) = (1/2)*[cos(x-y) + cos(x+y)]

sin(x)cos(y) = (1/2)*[sin(x+y) + sin(x+y)]

cos(x)sin(y) = (1/2)*[sin(x-y) – sin(x+y)]

Inverse Functions and Compositions

Inverse Function Relationships:

(remember that p = pi radians)

arcsin(x) + arccos(x) = p/2

arctan(x) + arccot(x) = p/2

arctan(x) + arctan(1/x) = p/2, if x>0

arctan(x) + arctan(1/x) = -p2, if x<0

Compositions of Trig Functions and Inverses:

sin(arccos(x)) = sqrt[1-x²]

tan(arcsin(x)) = x/ sqrt[1-x²]

sin(arctan(x)) = x/ sqrt[1+x²]

tan(arccos(x)) = sqrt[1-x²] / x

cos(arctan(x)) = 1/ sqrt[1+x²]

cot(arcsin(x)) = sqrt[1-x²] / x

cos(arcsin(x)) = sqrt[1-x²]

cot(arccos(x)) = x/ sqrt[1-x²]

The Exponential Function and the Trig Functions

The following definitions relate the trig functions you already know with the exponential function e and the natural logarithm function ln. This list assumes prior knowledge of the e and ln functions, and their relationships.

The imaginary number i is also used in this section. For review: i = sqrt(-1)

Exponential Definitions of Trig Functions

sin(x) = [e^ix – e^-ix] / [2i]

cos(x) = [e^ix + e^-ix] / [2i]

tan(x) = [e^ix – e^-ix] / [i*(e^ix + e^-ix)]

csc(x) = [2i] / [e^ix – e^-ix]

sec(x) = [2i] / [e^ix + e^-ix]

cot(x) = [I*(e^ix + e^-ix)] / [e^ix – e^-ix]

Exponential Definitions of Inverse Trig Functions

arcsin(x) = -i*ln(ix+sqrt[1-x²])

arccos(x) = -i*ln(x+sqrt[x²-1])

arctan(x) = (i/2)*ln([i+x]/[i-x])

arccsc(x) = -i*ln( (i/x) + sqrt[1-(1/x²)] )

arcsec(x) = -i*ln( (1/x) + sqrt[1-(i/x²)] )

arccot(x) = (i/2)*ln([x-i]/[x+i])

Trig Definition of the Exponential Function and Natural Log

cis(x) = e^ix = cos(x) + i*sin(x)

arccis(x) = (1/i)*ln(x)

Note: The cis(x) notation is not used by all teachers/professors, so check with yours about the best notation.

This post is part of the series: Trig Help

1. A Guide to Proving Trig Identities
2. Ace the Trig Exam With This Study Guide
3. Functions List for Trigonometry