(x sin a + y cos a)^2+ (x cos a – y sin a)^2 = x^2 + y^2

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