Final answer:
The sum sin9x + sin10x can be expressed as a product of sines and cosines using the sum-to-product identity, resulting in the simplified expression 2 sin(19x/2) cos(x/2).
Step-by-step explanation:
To express the sum sin9x + sin10x as a product of sines and/or cosines, we use the sum-to-product identities. Specifically, we apply the identity sin a + sin β = 2 sin((a + β)/2) cos((a - β)/2). For sin9x + sin10x, a is 9x and β is 10x.
The application of the identity gives us:
sin9x + sin10x = 2 sin((9x + 10x)/2) cos((9x - 10x)/2)
Simplifying the expressions within the sine and cosine functions:
sin9x + sin10x = 2 sin(19x/2) cos(-x/2)
Since cos(theta) is an even function, cos(-x/2) is the same as cos(x/2), so we further simplify to:
sin9x + sin10x = 2 sin(19x/2) cos(x/2)
This is the simplified expression for the given sum as a product of sines and cosines.