Download CHAPTER 14: PROPERTIES OF TRIGONOMETRIC

Section 14-2:
Trigonometric Identities
Objective
Given a trigonometric equation, prove that it is
an identity.
Purposes
There are two purposes for learning how to
prove identities.
To learn the relationships among the functions.
To learn to transform one trigonometric expression
into another equivalent form, usually by
simplifying it.
agreement
To prove that an equation is an identity, start
with one member and transform it into the
other.
Example
Prove:
1  cos x 1  cos x   sin
2
x
Notes
When working on identities, we must only
work on one side of the equal sign!!!
Example
Prove:
cot x  tan x  csc x sec x
Example
Prove:
sin x
1  cos x

1  cos x
sin x
Example
Prove:
csc  cos   sin   csc 
2
Steps in Proving Identities
Pick the member you wish to work with and write it down. Usually it is easier to
start with the more complicated member.
Look for algebraic things to do:
If there are two terms and you want only one:
Add fractions
Factor something out.
Multiply by a clever form of 1.
To multiply a numerator or denominator by its conjugate.
To get a desired expression in a numerator or denominator.
Do any obvious algebra such as:
Distributing
Squaring
Multiplying polynomials
Look for trigonometric things to do:
Look for familiar trigonometric expressions.
If there are squares of functions, think of the Pythagorean properties.
Keep looking at the answer to make sure you are headed in the right direction.
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