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51
Sum and Difference Identities
In the last lecture, we learned how to use basic identities to verify other interrelationships
between the trigonometric functions. In this lecture, we will append some additional identities
called the Sum and Difference Identities to our list of basic facts that we can use to verify further
interrelationships.
We want to establish an identity for the sine of the sum of two angles  and  .
Consider AB in the diagram below. Let AB revolve to point C and sweep out angle CAB ,
measuring  , then revolve further to point D and sweep out angle CAD , measuring  , as
shown.
B
From point D, we draw DE perpendicular to AB . Then, sin      ED DA , and
cos      AE DA .
Next, we draw DF perpendicular to AC , FG perpendicular to AB , and FH
perpendicular to ED . This makes AC a transversal crossing the parallels AB and HF .
Hence, HFA and CAB are alternate interior angles and congruent. Moreover, HDF
and HFA are also congruent because they are both complementary to HFD . We have
CAB  HFA and HDF  HFA . By the transitive property, CAB  HDF . Thus,
m  HDF    .
Now we note ED  GF  HD , and we substitute as below.
sin     
ED GF  HD GF HD



DA
DA
DA DA
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Multiplying by a judicious form of 1, we obtain the following.
GF HD

DA DA
GF AF HD FD
sin     



DA AF DA FD
GF AF HD FD
sin     



AF DA FD DA
sin     
Now, we note the following equalities by definition, sin    GF AF , cos     AF DA ,
cos    HD FD , and sin     FD DA . We conclude by substitution.
GF AF HD FD



AF DA FD DA
sin      sin    cos     cos    sin   
sin     
Hence, we have the following identity.
Sine of a Sum Identity:
sin      sin  cos   cos  sin 
The remaining Sum and Difference Identities we will state without proof.
Sum and Difference Identities:
sin      sin  cos   cos  sin 
sin      sin  cos   cos  sin 
cos      cos  cos   sin  sin 
cos      cos  cos   sin  sin 
tan     
tan   tan 
1  tan  tan 
tan     
tan   tan 
1  tan  tan 
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Example Exercise 1


Verify cos  x    cos x tan x .
2

By the Difference Identity of Cosine, we obtain the following.


cos  x    cos x tan x
2

 
 
cos  x  cos    sin  x  sin    cos x tan x
2
2
 
 
Evaluating cos   and sin   yields
2
2
cos  x   0  sin  x  1  cos x tan x
sin  x   cos x tan x.
Multiplying by a fortuitous form of “1” yields
cos x
 sin x  cos x tan x
cos x
sin x
cos x 
 cos x tan x.
cos x
The Quotient Identity completes the verification.
cos x  tan x  cos x tan x
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Example Exercise 2
Verify
cos  x  y  1  tan x tan y
.

sin  x  y  tan x  tan y
By the Quotient Identity, we have
sin x sin y
cos  x  y 
cos x cos y
.

sin x sin y
sin  x  y 

cos x cos y
1
Adding the fractions in the denominator, we obtain
sin x sin y
cos  x  y 
cos x cos y

sin  x  y  sin x  cos y sin y  cos x

cos x  cos y cos y  cos x
sin x sin y
1
cos  x  y 
cos x cos y

.
sin  x  y  sin x  cos y  sin y  cos x
cos x  cos y
1
Division, yields
cos  x  y 

sin x  sin y 
cos x  cos y
 1 

sin  x  y   cos x  cos y  sin x  cos y  sin y  cos x
cos  x  y 
cos x  cos y
sin x  sin y


.
sin  x  y  sin x  cos y  sin y  cos x sin x  cos y  sin y  cos x
Adding the fractions, we obtain
cos  x  y 
sin  x  y 

cos x  cos y  sin x  sin y
.
sin x  cos y  sin y  cos x
The Cosine of a Difference Identity and the Sine of a Sum Identity complete the verification.
cos  x  y 
sin  x  y 

cos  x  y 
sin  x  y 
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Suggested Homework in Dugopolski
Section 6.3: #11-17 odd, #79-97 odd
Suggested Homework in Ratti & McWaters
Section 6.2: #27-31 odd, #63-69 odd
Application Exercise
In the electromagnetic wave theory of light, scientists use the following equation
where k is the index of refraction, sin i sin r .
 k cos r  cos i 
E "  E 
 k cos r  cos i 
Show that the above equation is equivalent to the equation below.
 sin  i  r  
E "  E 

 sin  i  r  
Homework Problems
Use identities to find the exact value of the following expressions.
#1 sin  75 
#2 cos  12π 
#3 sin  1112π 
#4 cos  912π  cos  512π   sin  912π  sin  512π 
Verify the following identities.
#5 cos  α  180   cos α
#6 sin x  cos  32π  x 
#7 tan  x   tan  π  x 
#8 sin  360  x    sin x
#9 cos x cos y  12 cos  x  y   cos  x  y 
#10 tan x  tan  x  3π 
#11 cos 2 x  1  2sin 2 x
#12 sin 2 x  2sin x cos x