1. Product-to-Sum and Sum-to-Product Formulas

2. Product-to-Sum Formulas

3. Example
Express the following product as a sum or difference:
Solution

4. Text Example
Express each of the following products as a sum or difference.
a. sin 8x sin 3x b. sin 4x cos x
Solution The product-to-sum formula that we are using is shown in each of the voice balloons. a.
sin 8x sin 3x = 1/2[cos (8x - 3x) - cos(8x + 3x)] = 1/2(cos 5x - cos 11x)
sin  sin  = 1/2 [cos( - ) - cos( + )]
sin  cos  = 1/2[sin( + ) + sin( - )] b.
sin 4x cos x = 1/2[sin (4x + x) + sin(4x - x)] = 1/2(sin 5x + sin 3x)

5. Sum-to-Product Formulas

6. Example
Express the difference as a product:
Solution

7. Example
Express the sum as a product:
Solution

8. Example
Verify the following identity:
Solution