Precalculus
Basic Trigonometric Identities
Just as factoring helps us to simplify algebraic expressions, trigonometric relations help us to simplify and
rewrite trigonometric expressions. When trigonometry is applied to real world applications, additions,
subtractions, multiplications, and divisions can lead to rather imposing expressions. Trigonometric
identities help us reduce these complex expressions to more convenient form. As you work with
trigonometric identities, you will need to apply many of the skills of manipulation for algebraic expressions
you learned in algebra.
Identities are specials kinds of equations. An identity is an equation that is true for all valid replacements of
the variable.
example:
x + x = 2x
true for all x
x
x
true for all x, except when x = 0
=1
The Fundamental Trigonometric Identities
Reciprocal Identities
1
1.
sin θ =
csc θ
1
2.
csc θ =
sin θ
1
3.
cos θ =
sec θ
1
4.
sec θ =
cos θ
1
5.
tan θ =
cot θ
1
6.
cot θ =
tan θ
Product Identities
7.
sin θ • csc θ = 1
8.
cos θ • sec θ = 1
9.
tan θ • cot θ = 1
Pythagorean Identities
12.
cos 2 θ + sin2 θ = 1
sin2 θ = 1 − cos 2 θ
cos 2 θ = 1 − sin2 θ
13.
1 + tan2 θ = sec 2 θ
1 = sec 2 θ − tan2 θ
tan2 θ = sec 2 θ − 1
14.
cot 2 θ + 1 = csc 2 θ
1 = csc 2 θ − cot 2 θ
cot 2 θ = csc 2 θ − 1
Ratio Identities
10.
11.
sin θ
cos θ
cos θ
cot θ =
sin θ
tan θ =