# Trig Identities

Identities are equations that are always true, no matter what the value of the variables. These basic identities are used in analytic trig and calculus.

# Reciprocal Trig Identities

The reciprocal identities are derived from the definitions of the 3 basic trig functions. They are based on the definition of three of the functions as reciprocals of the basic three - sine, cosine, and tangent.

For any angle θ:

 sin θ = 1 csc θ
 csc θ = 1 sin θ
 cos θ = 1 sec θ
 sec θ = 1 cos θ
 tan θ = 1 cot θ
 cot θ = 1 tan θ

# Quotient Trig Identities

The quotient identities are derived from the definitions of the 3 basic trig functions in a unit circle and substitution from the definitions.

In a unit circle, sin = y, cos = x, and tan = y/x. Using substitution, for any angle θ:

 tan θ = sin θ cos θ
 cot θ = cos θ sin θ

# Pythagorean Trig Identities

The pythagorean identities are based on the Pythagorean Theorem, which states that in a right triangle with legs a and b and hypotenuse c:

a2 + b2 = c2

Using substitution into the Phythagorean Theorem from the trig definitions, for any angle θ:

sin2 + cos2 = 1

From this identity, also called the Fundamental Theorem of Trig, can be derived the other two related identities:

1 + tan2 = sec2

cot2 + 12 = csc2

Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 1/7/2011