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Basic Trigonometric Identities
•
Section 3.1
69
Example 3.5
Prove that
tan
2
θ
+
2
1
+
tan
2
θ
=
1
+
cos
2
θ
.
Solution:
Expand the left side:
tan
2
θ
+
2
1
+
tan
2
θ
=
(
tan
2
θ
+
1
)
+
1
1
+
tan
2
θ
=
sec
2
θ
+
1
sec
2
θ
(by (3.10))
=
sec
2
θ
sec
2
θ
+
1
sec
2
θ
=
1
+
cos
2
θ
When trying to prove an identity where at least one side is a ratio of expressions,
cross
multiplying
can be an effective technique:
a
b
=
c
d
if and only if
ad
=
bc
Example 3.6
Prove that
1
+
sin
θ
cos
θ
=
cos
θ
1
−
sin
θ
.
Solution:
Crossmultiply and reduce both sides until it is clear that they are equal:
(1
+
sin
θ
)(1
−
sin
θ
)
=
cos
θ
·
cos
θ
1
−
sin
2
θ
=
cos
2
θ
By (3.5) both sides of the last equation are indeed equal. Thus, the original identity holds.
Example 3.7
Suppose that
a
cos
θ
=
b
and
c
sin
θ
=
d
for some angle
θ
and some constants
a
,
b
,
c
, and
d
. Show
that
a
2
c
2
=
b
2
c
2
+
a
2
d
2
.
Solution:
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Trigonometric Identities

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