We have seen the usefulness of the Pythagorean Trigonometric Identities. If we combine those identities with other trigonometric identities, we have a powerful set of tools to help simplify expressions, factor expressions, and solve equations involving trigonometric functions.

We know that in mathematics, being able to look at equivalent "forms" of an expression or equation often sheds light on how to solve a problem. This toolbox of identities will help us look at problems from a different (but equivalent) viewpoint to reach a solution.

Trigonometric Identities Toolkit

Reciprocals Quotients Pythagorean
iden37 iden38 iden39
iden40 iden41 iden42
iden43 toolbox iden44

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Suggestions to Keep in Mind:
• When simplifying expressions, or factoring expressions,
look for common factors.
• If none exist, look for related functions from the identities
that will assist in the simplification.

 

ex1
Simplify:    cot θ sec θ   in terms of   sin θ
 
Solution:
The objective is to replace cot and sec with expressions that contain sin, so as to introduce sin into the expression.
more47

Substitutions from toolkit:
iden41

more46

 
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ex2
Simplify:    more444
 

Solution:
Look carefully! The numerator and the denominator are "versions" of the Pythagorean Identities.more49

Pythagorean Identities:
1 - sin2x = cos2x
csc2x - 1 = cot2x

Substitute:
more41

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ex3
Factor and simplify:  sec4x - tan4x
 

Solution:
This problem is the difference of perfect squares. Start by factoring. Use a Pythagorean Identity for substitution.
more52
more53

Factor difference of perfect squares.

Substitute:
tan2x + 1 = sec2x


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ex4
Multiply and express as a monomial:
(3 - 3sinx)(3 + 3sinx)
 

Solution:
Expand (multiply) first, then cancel and substitute.
more56

Substitute:
1 - sin2x = cos2x


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ti84c
For help with verifying trig identities on
your calculator,
click here.
MathBits Calculator Pages


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