To use the product-to-sum formula to fill in the blanks, we need to express sin(9x) cos(6x) in terms of sums of sines.
The product-to-sum formula states that sin(a)cos(b) = (1/2)[sin(a + b) + sin(a - b)].
Using this formula, we can rewrite the given expression as:
sin(9x) cos(6x) = (1/2)[sin(9x + 6x) + sin(9x - 6x)]
Simplifying the angles inside the sine functions, we get:
sin(9x) cos(6x) = (1/2)[sin(15x) + sin(3x)]
So, the blanks in the identity are filled as follows:
sin(9x) cos(6x) = ½(sin 15x + sin 3x)