Prove the product formula for sin x + sin y.

Expert Answers
justaguide eNotes educator| Certified Educator

The sum to product formula for sin x + sin y is `2*sin((x + y)/2)*cos((x-y)/2)`. This can be proved in the following way.

Use the formula for sine of the sum and difference of two values:

sin(A+B) = sin A*cos B+cos A*sin B ...(1)

sin(A - B) = sin A*cos B - cos A*sin B ...(2)

Add (1) and (2)

=> sin (A + B) + sin (A - B) = 2*sin A*cos B

substitute x = A + B and y = A - B which gives `A = (x + y)/2` and `B = (x - y)/2`

  `sin (A + B) + sin (A - B) = 2*sin A*cos B`

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

This proves the sum to product formula for `sin x + siny = 2*sin(x + y)/2*cos(x - y)/2`