How do you use the sum to product formulas to write the sum or difference sin(alpha+beta)-sin(alpha-beta)sin(α+β)sin(αβ) as a product?

1 Answer
Jan 13, 2017

The answer is =2cosalphasinbeta=2cosαsinβ

Explanation:

sin(A+-B)=sinAcosB+-cosAsinBsin(A±B)=sinAcosB±cosAsinB

Therefore

sin(alpha+beta)=sinalphacosbeta+cosalphasinbetasin(α+β)=sinαcosβ+cosαsinβ

sin(alpha-beta)=sinalphacosbeta-cosalphasinbetasin(αβ)=sinαcosβcosαsinβ

sin(alpha+beta)-sin(alpha-beta)=(sinalphacosbeta+cosalphasinbeta)-(sinalphacosbeta-cosalphasinbeta)sin(α+β)sin(αβ)=(sinαcosβ+cosαsinβ)(sinαcosβcosαsinβ)

=cancel(sinalphacosbeta)+cosalphasinbeta-cancel(sinalphacosbeta)+cosalphasinbeta

=2cosalphasinbeta