How do I write this product as a sum?

harry-p

cosxsin4x

puggy

cos(x) sin(4x)

Use the double angle identity.

sin(2y) = 2sin(y)cos(y), so

sin(4x) = sin(2*2x) = 2sin(2x)cos(2x).

But sin(2x) = 2sin(x)cos(x), and cos(2x) = cos^2(x) - sin^2(x), so

sin(4x) = 2[2sin(x)cos(x)] [cos^2(x) - sin^2(x)]

sin(4x) = 4sin(x)cos(x) [cos^2(x) - sin^2(x)]

sin(4x) = 4sin(x)cos^2(x) - 4sin^3(x)cos(x)

cos(x)sin(4x) = cos(x) [ 4sin(x)cos^2(x) - 4sin^3(x)cos(x) ]

= 4cos^3(x) sin(x) - 4sin^3(x)cos^2(x)