Analytic Trigonometry : The Sum-to-Product Formulas

How do we write the sum or difference of sines and/or cosines as products? We use
the following identities, which are called the sum-to-product formulas:
Sum-to-Product Formulas
sin a sin b = 2 sin a b
2 cos a – b
2
sin a – sin b = 2 sin a – b
2 cos a b
2
cos a cos b = 2 cos a b
2 cos a – b
2
cos a – cos b = -2 sin a b
2 sin a – b
2
We verify these formulas using the product-to-sum formulas. Let’s verify the first
sum-to-product formula
sin a sin b = 2 sin a b
2 cos a – b
2 .
We start with the right side of the formula, the side with the product. We can apply
the product-to-sum formula for sin A cos B to this expression. By doing so, we obtain
the left side of the formula, sin a sin b. Here’s how:
The three other sum-to-product formulas in the preceding box are verified in a
similar manner. Start with the right side and obtain the left side using an appropriate
product-to-sum formula.
Example 2 Using the Sum-to-Product Formulas
Express each sum or difference as a product:
a. sin 9x sin 5x b. cos 4x – cos 3x.
Solution
The sum-to-product formula that we are using is shown in each of the voice balloons.
a.
9x 5x
2
9x − 5x
2 sin 9x sin 5x = 2 sin cos
14x
2
4x
2 = 2 sin cos = 2 sin 7x cos 2x
UKPa UKPb =UKP EQU a b

a – b

b.
4x 3x
2
4x − 3x
2 cos 4x − cos 3x = –2 sin sin
EQUa –EQUb = –UKP UKP a b

a – b

7x
2
x
2 = –2 sin