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Algebra and trigonometry
Trigonometric identities and
Solving trigonometric equations
Reciprocal Identities
sin
θ
=
1
csc
θ
csc
θ
=
1
sin
θ
cos
θ
=
1
sec
θ
sec
θ
=
1
cos
θ
tan
θ
=
1
cot
θ
cot
θ
=
1
tan
θ
The final set of identities is the set of
quotient identities , which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See
[link] .
Quotient Identities
tan
θ
=
sin
θ
cos
θ
cot
θ
=
cos
θ
sin
θ
The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.
A General Note
Summarizing trigonometric identities
The
Pythagorean identities are based on the properties of a right triangle.
cos
2
θ
+
sin
2
θ
=
1
1
+
cot
2
θ
=
csc
2
θ
1
+
tan
2
θ
=
sec
2
θ
The
even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle.
tan
(
−
θ
)
=
−
tan
θ
cot
(
−
θ
)
=
−
cot
θ
sin
(
−
θ
)
=
−
sin
θ
csc
(
−
θ
)
=
−
csc
θ
cos
(
−
θ
)
=
cos
θ
sec
(
−
θ
)
=
sec
θ
The
reciprocal identities define reciprocals of the trigonometric functions.
sin
θ
=
1
csc
θ
cos
θ
=
1
sec
θ
tan
θ
=
1
cot
θ
csc
θ
=
1
sin
θ
sec
θ
=
1
cos
θ
cot
θ
=
1
tan
θ
The
quotient identities define the relationship among the trigonometric functions.
tan
θ
=
sin
θ
cos
θ
cot
θ
=
cos
θ
sin
θ
How To
Given a trigonometric identity, verify that it is true.
Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
Look for opportunities to factor expressions, square a binomial, or add fractions.
Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
If these steps do not yield the desired result, try converting all terms to sines and cosines.
Try It
Verify the identity
csc
θ
cos
θ
tan
θ
=
1.
csc
θ
cos
θ
tan
θ
=
(
1
sin
θ
)
cos
θ
(
sin
θ
cos
θ
)
=
cos
θ
sin
θ
(
sin
θ
cos
θ
)
=
sin
θ
cos
θ
sin
θ
cos
θ
=
1
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Verifying the equivalency using the even-odd identities
Verify the following equivalency using the even-odd identities:
(
1
+
sin
x
)
[
1
+
sin
(
−
x
)
]
=
cos
2
x
Working on the left side of the equation, we have
(
1
+
sin
x
)
[
1
+
sin
(
−
x
)
]
=
(
1
+
sin
x
)
(
1
−
sin
x
)
Since sin(−
x
)=
−
sin
x
=
1
−
sin
2
x
Difference of squares
=
cos
2
x
cos
2
x
=
1
−
sin
2
x
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Verifying a trigonometric identity involving
sec
2 θ
Verify the identity
sec
2
θ
−
1
sec
2
θ
=
sin
2
θ
As the left side is more complicated, let’s begin there.
sec
2
θ
−
1
sec
2
θ
=
(
tan
2
θ
+
1
)
−
1
sec
2
θ
sec
2
θ
=
tan
2
θ
+
1
=
tan
2
θ
sec
2
θ
=
tan
2
θ
(
1
sec
2
θ
)
=
tan
2
θ
(
cos
2
θ
)
cos
2
θ
=
1
sec
2
θ
=
(
sin
2
θ
cos
2
θ
)
(
cos
2
θ
)
tan
2
θ
=
sin
2
θ
cos
2
θ
=
(
sin
2
θ
cos
2
θ
)
(
cos
2
θ
)
=
sin
2
θ
There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side.
sec
2
θ
−
1
sec
2
θ
=
sec
2
θ
sec
2
θ
−
1
sec
2
θ
=
1
−
cos
2
θ
=
sin
2
θ
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Creating and verifying an identity
Create an identity for the expression
2
tan
θ
sec
θ
by rewriting strictly in terms of sine.
There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression:
2
tan
θ
sec
θ
=
2
(
sin
θ
cos
θ
)
(
1
cos
θ
)
=
2
sin
θ
cos
2
θ
=
2
sin
θ
1
−
sin
2
θ
Substitute
1
−
sin
2
θ
for
cos
2
θ
.
Thus,
2
tan
θ
sec
θ
=
2
sin
θ
1
−
sin
2
θ
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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