Express the following as a product containing only sines and/or cosines.
sin9x−sin7x
To express the expression sin(9x) - sin(7x) as a product containing only sines and/or cosines, we can use the trigonometric identity for the difference of two sines:
sin(a) - sin(b) = 2 * cos((a+b)/2) * sin((a-b)/2)
Applying this identity, we can rewrite sin(9x) - sin(7x) as:
2 * cos((9x+7x)/2) * sin((9x-7x)/2)
Simplifying the angles, we get:
2 * cos((16x)/2) * sin((2x)/2)
Further simplifying, we have:
2 * cos(8x) * sin(x)
So, sin(9x) - sin(7x) can be expressed as the product:
2 * cos(8x) * sin(x)
2*sin[(1/2)*(9*x - 7*x)]*cos[(1/2)*(9*x + 7*x)]
2*sin[x]*cos[8*x]