1. MATH 107
Section 6.2
Sum and Difference Formulas

2. 2© 2011 Pearson Education, Inc. All rights reserved
SUM AND DIFFERENCE FORMULAS
FOR COSINE
( )
( )
cos cos cos sin sin
cos cos cos sin sin
u v u v u v
u v u v u v
+ = −
− = +

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EXAMPLE 1 Using the Difference Identity for Cosine
Find the exact value of by using
π
12
=
π
3
−
π
4
.
Solution
cos cos
12 3 4
π π π 
= − ÷
 
=
1
2
×
2
2
+
3
2
×
2
2
=
2
4
+
6
4
=
2 + 6
4
= cos
π
3
cos
π
4
+ sin
π
3
sin
π
4
12
cos
π

4. Example
4© 2011 Pearson Education, Inc. All rights reserved
Find the exact value of cos15 .°

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SUM AND DIFFERENCE IDENTITIES
FOR SINE
( )
( ) vuvuvu
vuvuvu
sincoscossinsin
sincoscossinsin
−=−
+=+

6. 19
Find the exact value of sin .
12
π
(Answer on next slide.)

7. 19
Find the exact value of sin .
12
π
19 16 3 4
sin sin sin
12 12 12 3 4
π π π π π   
= + = + ÷  ÷
   

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EXAMPLE 5 Using the Sum Formula for Sine
Find the exact value of
sin63º cos27º + cos63ºsin27º
without using a calculator.
( )sin63ºcos27º cos63ºsin27º sin 63º 27º+ = +
( )sin 90º=
= 1
Solution
This expression is the right side of the sum identity
for sin (u + v), where u = 63º and v = 27º.

9. Example
9© 2011 Pearson Education, Inc. All rights reserved
Find the exact value of sin75°

10. 10© 2011 Pearson Education, Inc. All rights reserved
.sin
2
cos vv =





−
π
BASIC COFUNCTION IDENTITIES
If v is any real number or angle measured in
radians, then
If angle v is measured in degrees, then
replace by 90º in these identities.
π
2
.cos
2
sin vv =





−
π

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EXAMPLE 3 Using Cofunction Identities
Prove that for any real number x, tan cot .
2
x x
π 
− = ÷
 
Solution
tan
π
2
− x



÷ =
sin
π
2
− x



÷
cos
π
2
− x



÷
cos
sin
x
x
=
cot x=

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SUM AND DIFFERENCE IDENTITIES
FOR TANGENT
( )
( )
tan tan
tan
1 tan tan
tan tan
tan
1 tan tan
u v
u v
u v
u v
u v
u v
−
− =
+
+
+ =
−

13. 13© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 11 Verifying an Identity
Verify the identity .tan π − x( ) = − tan x
Solution
Apply the difference identity.
( )
tan tan
tan
1 tan tan
x
x
x
π
π
π
−
− =
+
( )
0 tan
tan
1 0 tan
x
x
x
π
−
− =
+ ×
( )tan tanx xπ − = −
Therefore, the given equation is an identity.