Answer:
The identity sin (x + y) - sin (x - y) = 2 cos x sin y is verified
Step-by-step explanation:
Let us revise com rules in trigonometry
sin(Ф + α) = sinФ cosα + cosФ sin α
sin(Ф - α) = sinФ cos α - cosФ sinα
To verify the identity sin (x + y) - sin (x - y) = 2 cos x sin y, take the left hand side and simplify it to give the right hand side
∵ L.H.S = sin (x + y) - sin (x - y)
∵ sin (x + y) = sin x cos y + cos x sin y
∵ sin (x - y) = sin x cos y - cos x sin y
- Substitute then in the left hand side
∴ L.H.S = [sin x cos y + cos x sin y] - [sin x cos y - cos x sin y]
- simplify it and remember (-)(-) = (+)
∴ L.H.S = sin x cos y + cos x sin y - sin x cos y + cos x sin y
- Add the like terms
∵ sin x cos y - sin x cos y = 0
∵ cos x sin y + cos x sin y = 2 cos x sin y
∴ L.H.S = 2 cos x sin y
∵ R.H.S = 2 cos x sin y
∴ L.H.S = R.H.S
The identity sin (x + y) - sin (x - y) = 2 cos x sin y is verified
