Question

write 3sin2xsin3x as a sum or difference

Answer 1

sum formula for sin/cos: sin2x=2sinxcosx, and cos2x=cos^2x-sin^2x
notice that sin3x=sin(x+2x)=sinxcos2x+sin2xcosx=sinx(cos^2x-sin^2x)+2sinxcosx*cosx
NOTE: sin^2x is my notation for (sinx)^2 and similarly for cosine.

so in your original problem,
3(sin2x)(sin3x)
=3 (2sinx cosx) (sinx (cos^2x - sin^2x) + 2sinx cosx cosx)
=6 sinx cosx (sinx cos^2x-sin^3x + 2 sinx cos^2x)
=6 sin^2x cos^3x - 6sin^4x cosx + 12 sin^2x cos^3x