Question

How do you use sum to product formulas to write the sum or difference `sin(x+pi/4)-sin(x-pi/4)` as a product?

Answer 1

`sin(x+pi/4)-sin(x-pi/4)=sqrt2cosx`

Explanation:

As `sin(A+B)=sinAcosB+cosAsinB` and

`sin(A-B)=sinAcosB-cosAsinB`

subtracting second from first we get `sin(A+B)-sin(A-B)=2cosAsinB`

Hence `sin(x+pi/4)-sin(x-pi/4)=2cosxsin(pi/4)`

= `2cosx xx1/sqrt2=sqrt2cosx`