Arkansas Tech University
MATH 1203: Trigonometry
Dr. Marcel B. Finan
18
Verifying Trigonometric Identities
In this section, you will learn how to use trigonometric identities to simplify
trigonometric expressions.
Equations such as
x2 − 1
=x+1
x−1
are referred to as identities. An identity is an equation that is true for
all values of x for which the expressions in the equation are defined. For
example, the equation
(x − 2)(x + 2) = x2 − 4 or
(x − 2)(x + 2) = x2 − 4
is defined for all real numbers x. The equation
x2 − 1
=x+1
x−1
is true for all real numbers x 6= 1.
We have already seen many trigonometric identities. For the sake of completeness we list these basic identities:
Reciprocal Identities
sin x =
csc x =
tan x =
1
csc x
1
sin x
1
cot x
cos x =
sec x =
tan x =
1
sec x
1
cos x
1
cot x
Quotient identities
tan t =
sin t
cos t
; cot t =
cos t
sin t
Pythagorean identities
cos2 x + sin2 x =
1
1 + tan2 x
= sec2 x
1 + cot2 x
= csc2 x
1
Even-Odd identities
sin (−x) = − sin x cos (−x) =
cos x
csc (−x) = − csc x sec (−x) =
sec x
tan (−x) = − tan x cot (−x) = − cot x
Simplifying Trigonometric Expressions
Some algebraic expressions can be written in different ways. Rewriting a
complicated expression in a much simpler form is known as simplifying the
expression. There are no standard steps to take to simplify a trigonometric expression. Simplifying trigonometric expressions is similar to factoring
polynomials: by trial and error and by experience, you learn what will work
in which situations. To simplify algebraic expressions we used factoring,
common denominators, and other formulas. We use the same techniques
with trigonometric expressions together with the fundamental trigonometric
identities listed above.
Example 18.1
Simplify the expression
sec2 θ−1
.
sec2 θ
Solution.
Using the identity 1 + tan2 θ = sec2 θ we find
sec2 θ − 1 1 + tan2 θ − 1
=
sec2 θ
sec2 θ
tan2 θ
= 2
sec θ
sin2 θ
= 2 cos2 θ = sin2 θ
cos θ
Example 18.2
Simplify the expression:
sin θ
1+cos θ
+
1+cos θ
.
sin θ
Solution.
Taking common denominator and using the identity cos2 θ + sin2 θ = 1 we
find
2
sin θ
1 + cos θ (1 + cos θ)2 + sin2 θ
+
=
1 + cos θ
sin θ
sin θ(1 + cos θ)
2(1 + cos θ)
=
sin θ(1 + cos θ)
=2 csc θ
Example 18.3
Simplify the expression: (sin x − cos x)(sin x + cos x).
Solution.
Multiplying we find
(sin x − cos x)(sin x + cos x) = sin2 x − cos2 x
Example 18.4
Simplify cos x + tan x sin x.
Solution.
Using the quotient identity tan x =
sin2 x = 1 we find
sin x
cos x
and the Pythagorean identity cos2 x+
sin x
sin x
cos x
cos2 x + sin2 x
=
cos x
1
=
= sec x.
cos x
cos x + tan x sin x = cos x +
Establishing Trigonometric Identities
A trigonometric identity is a trigonometric equation that is valid for all values
of the variable for which the expressions in the equation are defined. How
do you show that a trigonometric equation is not an identity? All you need
to do is to show that the equation does not hold for just one value of the
variable. For example, the equation
sin x + cos x = 1
3
π
4
we have
√
√
π
π
2
2 √
sin + cos =
+
= 2 6= 1.
4
4
2
2
is not an identity since for x =
To verify that an equation is an identity, we start by simplifying one side of
the equation and end up with the other side.
One of the common methods for establishing trigonometric identities is to
start with the side containing the more complicated expression and, using
appropriate basic identities and algebraic manipulations, such as taking a
common denominator, factoring and multiplying by a conjugate, to arrive at
the other side of the equality.
Example 18.5
Establish the identity:
1+sec θ
sec θ
=
sin2 θ
.
1−cos θ
Solution.
Using the identity cos2 θ + sin2 θ = 1 we have
1 − cos2 θ
sin2 θ
=
1 − cos θ 1 − cos θ
(1 − cos θ)(1 + cos θ)
=
1 − cos θ
=1 + cos θ = cos θ(1 + sec θ)
1 + sec θ
=
sec θ
Example 18.6
Show that sin θ = cos θ is not an identity.
Solution.
Letting θ =
π
2
we get 1 = sin π2 6= cos π2 = 0.
Example 18.7
Verify the identity: cos x(sec x − cos x) = sin2 x.
Solution.
The left-hand side looks more complex then the right-hand side, so we start
with it and try to transform it to the right-hand side.
4
cos x(sec x − cos x) = cos x sec x − cos2 x
1
= cos x
= cos2 x
cos x
=1 − cos2 x = sin2 x
Example 18.8
Verify the identity: 2 tan x sec x =
1
1−sin x
−
1
.
1+sin x
Solution.
Starting from the right-hand side to obtain
1
1
(1 + sin x) − (1 − sin x)
2 sin x
−
=
=
1 − sin x 1 + sin x
(1 − sin x)(1 + sin x)
1 − sin2 x
=
Example 18.9
Verify the identity:
cos x
1−sin x
2 sin x
sin x 1
=2
= 2 tan x sec x
2
cos x
cos x cos x
= sec x + tan x.
Solution.
Using the conjugate of 1 − sin x to obtain
cos x(1 + sin x)
cos x
=
1 − sin x (1 − sin x)(1 + sin x)
=
cos x + cos x sin x
1 − sin2 x
=
cos x + cos x sin x
cos2 x
=
1
sin x
+
= sec x + tan x.
cos x cos x
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