Question

How do you use product to sum formulas to write the product `4sin3alphacos2alpha` as a sum or difference?

Answer 1

`4sin3alphacos2alpha=2sin5alpha+2sinalpha`

Explanation:

We know `sin(A+B)=sinAcosB+cosAsinB`
and `sin(A-B)=sinAcosB-cosAsinB`

Adding two `2sinAcosB=sin(A+B)=sin(A-B)`

Hence `4sin3alphacos2alpha`

= `2xx2sin3alphacos2alpha`

= `2xx(sin(3alpha+2alpha)+sin(3alpha-2alpha))`

= `2sin5alpha+2sinalpha`