To fill in the blanks using the product to sum formula, we can express the right side of the equation as a sum of two sine functions. The product to sum formula states that:
sin(A)cos(B) = 1/2 [sin(A + B) + sin(A - B)]
In this case, we have sin(9x)cos(8x) on the left side, so we can rewrite it using the formula:
sin(9x)cos(8x) = 1/2 [sin(9x + 8x) + sin(9x - 8x)]
Simplifying the expressions inside the brackets, we get:
sin(9x + 8x) = sin(17x)
sin(9x - 8x) = sin(x)
Therefore, the filled identity becomes:
sin(9x)cos(8x) = 1/2 [sin(17x) + sin(x)]
So, the blanks are filled as sin(17x) and sin(x).