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14.3 Verify Trigonometric Identities
Goal  Verify trigonometric identities.
Your Notes
VOCABULARY
Trigonometric identity
A trigonometric equation that is true for all values of  (in its domains)
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Reciprocal Identities
csc  =
1
sin
sec  =
1
cos
cot =
1
tan
Tangent and Cotangent Identities
tan  = sin 
cos 
cot  = cos 
sin 
Pythagorean Identities
sin2  + cos2  = __1__
1+ tan2  =__sec2 __
1+cot2  =__csc2__
Cofunction Identities
Negative Angle Identities
sin      =__cos __
2

sin () =__sin__
cos      =__sin __


2

cos () = __cos__
tan      =__cot __
2

tan () = __tan __
Your Notes
Example 1
Find trigonometric values
3
Given that cos  =  3 and  <  < , find the values of the other five trigonometric
2
4
functions of .
Solution
Step 1 Find sin .
sin 2  +cos 2 =1
Write Pythagorean Identity.
2
 3
sin2  +    =1
 4
3
for cos .
4
2
 3
Subtract    from each side.
 4
Substitute 
sin2 = 1   3 
 4
2
7
sin2  = _____
16
7
sin  =  4
Simplify.
Take square roots of each side.
Because  is in Quadrant __III__, sin  is negative. So, sin  = 
7
4.
Step 2 Find the values of the other four trigonometric functions of  using the known
values of sin  and cos .
7

tan  = sin   4  7
3
cos 
3

4
3
cot  =

cos 
4  3  3 7

sin 
7
7
7

csc  = 1
14
4
4 7

 


sin
7
7
 7

4
sec  = 1
1
4

 
cos   3
3
4

Your Notes
Checkpoint Find the values of the other five trigonometric functions of .
1

,0
4
2
sin  = 15 ; tan  = 15 ; cot  = 15 ;
4
15
csc  = 4 15 ; sec  = 4
1. cos  =
15
1 3
2. sin  =  ,
   2
3 2
cos  = 2 2 ; tan  = 2 2 ; cot  =  2 2 ;

3
8
csc  =  3; sec  = 6 2
8
Example 2
Simplify a trigonometric expression
1
 cot .

sin  0 
2

Solution
1  cot 
1
=
 cot  .
cos 


______
sin   0 
2

= 1  co 
ssin
cos  ______
_____
Simplify the expression

=
1

sin
_____
= __csc __
Checkpoint Simplify the expression.
3.
tan 
3
 sin  + tan  csc   cos 

sec
1
___Cofunction identity__
Cotangent identity
Simplify.
__Reciprocal identity__
Your Notes
Example 3
Verify a trigonometric identity
Verify the identity
1  cos 2 
1
sec  
 tan  cot  =
.
1  sin 2 
cos 
b
Solution
sec  
1
 tan  cot  = sec   __sec __  tan  cot 
cos 
= __sec2 _  tan  cot 
1
= ___sec2 __  tan 
tan 
_____
= __sec2 _  __1__
= __tan2 __
sin2 
=
_______
cos2 
2
= 1  cos 2 
1  sin 
Checkpoint Verify the identity.
4.
cos 2 (  x )
 sin 2 x
2
cot x
cos 2 ( x) cos2 x

cot 2 x
cot 2 x
cos 2 x
=
cos 2 x
sin 2 x
sin 2 x
 cos 2 x 
cos 2 x
= sin2 x
Homework
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