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.

The basic Trig Identities are divided into these types. Click on the link for more information.


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


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Last Updated: 1/7/2011
K. A. Shearer 2011
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