cos
(
α
−
β
)
=
cos
α
cos
β
+
sin
α
sin
β
−
cos
(
α
+
β
)
=
−
(
cos
α
cos
β
−
sin
α
sin
β
)
____________________________________________________
cos
(
α
−
β
)
−
cos
(
α
+
β
)
=
2
sin
α
sin
β
Then, we divide by 2 to isolate the product of sines:
sin
α
sin
β
=
1
2
[
cos
(
α
−
β
)
−
cos
(
α
+
β
)
]
Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.
A General Note
The
product-to-sum formulas are as follows:
cos
α
cos
β
=
1
2
[
cos
(
α
−
β
)
+
cos
(
α
+
β
)
]
sin
α
cos
β
=
1
2
[
sin
(
α
+
β
)
+
sin
(
α
−
β
)
]
sin
α
sin
β
=
1
2
[
cos
(
α
−
β
)
−
cos
(
α
+
β
)
]
cos
α
sin
β
=
1
2
[
sin
(
α
+
β
)
−
sin
(
α
−
β
)
]
Express the product as a sum or difference
Write
cos
(
3
θ
)
cos
(
5
θ
)
as a sum or difference.
We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.
cos
α
cos
β
=
1
2
[
cos
(
α
−
β
)
+
cos
(
α
+
β
)
]
cos
(
3
θ
)
cos
(
5
θ
)
=
1
2
[
cos
(
3
θ
−
5
θ
)
+
cos
(
3
θ
+
5
θ
)
]
=
1
2
[
cos
(
2
θ
)
+
cos
(
8
θ
)
]
Use even-odd identity
.
Got questions? Get instant answers now! Got questions? Get instant answers now!
Expressing sums as products
Some problems require the reverse of the process we just used. The
sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for
sine . Let
u
+
v
2
=
α
and
u
−
v
2
=
β
.
Then,
α
+
β
=
u
+
v
2
+
u
−
v
2
=
2
u
2
=
u
α
−
β
=
u
+
v
2
−
u
−
v
2
=
2
v
2
=
v
Thus, replacing
α
and
β
in the product-to-sum formula with the substitute expressions, we have
sin
α
cos
β
=
1
2
[
sin
(
α
+
β
)
+
sin
(
α
−
β
)
]
sin
(
u
+
v
2
)
cos
(
u
−
v
2
)
=
1
2
[
sin
u
+
sin
v
]
Substitute for
(
α
+
β
)
and
(
α
−
β
)
2
sin
(
u
+
v
2
)
cos
(
u
−
v
2
)
=
sin
u
+
sin
v
The other sum-to-product identities are derived similarly.
A General Note
The
sum-to-product formulas are as follows:
sin
α
+
sin
β
=
2
sin
(
α
+
β
2
)
cos
(
α
−
β
2
)
sin
α
−
sin
β
=
2
sin
(
α
−
β
2
)
cos
(
α
+
β
2
)
cos
α
−
cos
β
=
−2
sin
(
α
+
β
2
)
sin
(
α
−
β
2
)
cos
α
+
cos
β
=
2
cos
(
α
+
β
2
)
cos
(
α
−
β
2
)
Writing the difference of sines as a product
Write the following difference of sines expression as a product:
sin
(
4
θ
)
−
sin
(
2
θ
)
.
We begin by writing the formula for the difference of sines.
sin
α
−
sin
β
=
2
sin
(
α
−
β
2
)
cos
(
α
+
β
2
)
Substitute the values into the formula, and simplify.
sin
(
4
θ
)
−
sin
(
2
θ
)
=
2
sin
(
4
θ
−
2
θ
2
)
cos
(
4
θ
+
2
θ
2
)
=
2
sin
(
2
θ
2
)
cos
(
6
θ
2
)
=
2
sin
θ
cos
(
3
θ
)
Got questions? Get instant answers now! Got questions? Get instant answers now!
Evaluate
cos
(
15°
)
−
cos
(
75°
)
.
Check the answer with a graphing calculator.
We begin by writing the formula for the difference of cosines.
cos
α
−
cos
β
=
−
2
sin
(
α
+
β
2
)
sin
(
α
−
β
2
)
Then we substitute the given angles and simplify.
cos
(
15°
)
−
cos
(
75°
)
=
−2
sin
(
15°
+
75°
2
)
sin
(
15°
−
75°
2
)
=
−2
sin
(
45°
)
sin
(
−30°
)
=
−2
(
2
2
)
(
−
1
2
)
=
2
2
Got questions? Get instant answers now! Got questions? Get instant answers now!
Proving an identity
Prove the identity:
cos
(
4
t
)
−
cos
(
2
t
)
sin
(
4
t
)
+
sin
(
2
t
)
=
−
tan
t
We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.
cos
(
4
t
)
−
cos
(
2
t
)
sin
(
4
t
)
+
sin
(
2
t
)
=
−
2
sin
(
4
t
+
2
t
2
)
sin
(
4
t
−
2
t
2
)
2
sin
(
4
t
+
2
t
2
)
cos
(
4
t
−
2
t
2
)
=
−
2
sin
(
3
t
)
sin
t
2
sin
(
3
t
)
cos
t
=
−
2
sin
(
3
t
)
sin
t
2
sin
(
3
t
)
cos
t
=
−
sin
t
cos
t
=
−
tan
t
Got questions? Get instant answers now! Got questions? Get instant answers now!
