Sum and
Difference
Identities for
Cosine
Consider the equation
cos      cos   cos 
Is this an identity? Remember an identity means the
equation is true for every value of the variable for which
it is defined.
Let’s try  = 30° and β = 45°
?
cos  30  45   cos 30  cos 45
cos75  0.2588 cos30  cos 45  1.573
So cos  30  45  cos30  cos 45
This is NOT an identity and
DOES NOT WORK for all values!!!
Often you will have the cosine of the sum or difference
of two angles. We would like an identity to express
this in terms of products and sums of sines and
cosines. The proof of this identity is on Page 185-186
in your book. The identities are:
cos     cos cos   sin  sin 
cos     cos cos   sin  sin 
You will need to know these so say them in your
head when you write them like this, "The cosine of
the sum of 2 angles is cosine of the first, cosine of
the second minus sine of the first sine of the
second."



Find the exact value of cos 105 .
Since it says exact we want to use values we know from our
unit circle. 105° is not one there but can we take the sum or
difference of two angles from unit circle and get 105° ?
  cos60
cos 105
We can use the
sum formula and
get cosine of the
first, cosine of the
second minus
sine of the first,
sine of the
second.
 45

 cos 60 cos 45  sin 60 sin 45
1
2
3
2
 


2 2
2
2
2
6
2 6



4
4
4
The sum of all of the angles in a triangle always is 180°
What is the sum of  + ? 90°
Since we have a 90° angle, the sum of the other two angles
must also be 90° (since the sum of all three is 180°).

Two angles whose sum is
90° are called
complementary angles.
c
b
a
a
What is sin  ? c
a
What is cos ?
c

Since  and  are
complementary angles and
sin  = cos ,
sine and cosine are called
cofunctions.
This is where we get the name
cosine, a cofunction of sine.
Looking at the names of the other trig functions can
you guess which ones are cofunctions of each other?
secant and cosecant
tangent and cotangent
Let's see if this is right. Does sec  = csc ?
hypotenuse over adjacent

c
b
a

hypotenuse over
opposite
c
sec    csc 
b
This whole idea of the
relationship between
cofunctions can be
stated as:
Cofunctions of complementary
angles are equal.
Cofunctions of complementary angles are equal.
cos 27° = sin(90° - 27°) = sin 63°
Using the theorem above, what trig function of
what angle does this equal?
  cot       cot 3 


 
tan
8
2 8
 8 
Let's try one in radians. What trig functions of
what angle does this equal?

The sum of complementary angles in radians is
2
90° is the same as 
2

Basically any trig function then equals 90° minus or 2
minus its cofunction.
since
Cofunction Identities




sin   u   cosu cos   u   sin u
2

2





tan   u   cot u cot   u   tan u
2

2





sec   u   cscu csc   u   secu
2

2

sin 36 sin 36

 tan 36
sin 54 cos 36
We can't use fundamental identities if the trig functions are
of different angles.
Use the cofunction theorem to change the denominator
to its cofunction
Now that the angles are the same we can use a trig
identity to simplify.