Using the identity sin(2x) = 2sin(x)cos(x), we can express sin(5x) as:
sin(5x) = sin(3x + 2x) = sin(3x)cos(2x) + cos(3x)sin(2x)
Expanding sin(3x) using the identity sin(3x) = 3sin(x) - 4sin^3(x) and sin(2x) using the identity sin(2x) = 2sin(x)cos(x), we get:
sin(5x) = (3sin(x) - 4sin^3(x))(1 - 2sin^2(x)) + cos(3x)2sin(x)cos(x)
Simplifying, we get:
sin(5x) = 3sin(x) - 4sin^3(x) - 6sin(x)sin^2(x) + 2cos(x)cos(3x)sin(x)
Using the identity cos(3x) = 4cos^3(x) - 3cos(x), we get:
sin(5x) = 3sin(x) - 4sin^3(x) - 6sin(x)sin^2(x) + 2cos(x)(4cos^3(x) - 3cos(x))sin(x)
Simplifying further, we get:
sin(5x) = 3sin(x) - 4sin^3(x) - 6sin(x)sin^2(x) + 8cos^4(x)sin(x) - 6cos^2(x)sin(x)
Finally, factoring out sin(x), we get:
sin(5x) = sin(x)(3 - 4sin^2(x) - 6sin(x) + 8cos^4(x) - 6cos^2(x))
(If this doesn’t seem right then comment!)