Notes: Simplifying Trigonometric Expressions
An identity is an equation that is true for every number in the domain of the equation. An
identity that you already know is tan x =
sin x
. No matter what value you use for x, it is
cos x
ALWAYS true. We will be using identities and algebra to simplify trigonometric expressions.
That means ideally the new expression contains no fractions and contains the least number of
terms possible. First we need to review some identities you already know and learn a few new
ones. You need to MEMORIZE these!
Ones you already know…
Reciprocal Identities:
sin x =
1
csc x
cos x =
1
sec x
tan x =
1
cot x
csc x =
1
sin x
sec x =
1
cos x
cot x =
1
tan x
Ratio Identities:
tan x =
sin x
cos x
cot x =
cos x
sin x
Now let’s derive some new ones…
Since by the Pythagorean theorem, a2 + b2 = c2 that means by
substitution
This is commonly written as
Now, we can also create two new identities by dividing that
one by either sin2x or cos2x
Pythagorean Identities:
sin 2 x + cos 2 x = 1
1 + tan 2 x = sec2 x
cot 2 x + 1 = csc 2 x
Simplifying Trigonometric Expressions can be done a variety of ways. Even the same
expression could be simplified a variety of ways. Here are a few different methods…
Rewriting in terms of sine/cosine
Multiplying
tan x
=
sec x
(1 − cos x)(1 + sec x)(cos x)
Using a Pythagorean Identity
tan x(csc2 x − 1) =
Factoring
Factoring
sec2 x − 1
=
sec x − 1
cos x − cos x sin 2 x
Splitting a Fraction into Two Fractions
Getting a Common Denominator
sec x − cos x
sec x
sin x + cos x cot x
Miscellaneous
cos x
+ tan x
1 + sin x