Remember the rules of logarithms: the sum of the logarithms having the same base becomes the logarithm of the product of the numbers.

sin x + sin y = `2 sin (x+y)/2 *cos(x-y)/2`

sin x - sin y = `2 cos(x+y)/2 *sin(x-y)/2`

Use the notation `(x+y)/2 =alpha =gt x+y = 2 alpha`

`(x-y)/2 = beta =gt x-y = 2 beta`

Adding the equations yields: `2x = 2(alpha + beta) =gt x = alpha + beta`

Subtract x-y from x+y => `2y = 2(alpha + beta) =gt y = alpha - beta`

`sin (alpha + beta) + sin (alpha- beta) = sin alpha*cos beta + sin beta*cos alpha + sin alpha*cos beta - sin beta*cos alpha`

Reducing the opposite terms yields:

`sin (alpha + beta) + sin (alpha- beta) = 2 sin alpha*cos beta`