How do you prove this identity by working from the left side?

Question

sin (a+2) - sin(a) = 2 cos(a+1)sin(1)

Answer 1

sin(a+2) - sin(a)

=>sin[(a+1)+1]- sin[(a+1)-1]

use formulae for sin(a+b) and sin(a-b)

here a = a+ 1 and b= 1


sin(a+1)cos(1)+cos(a+1)sin(1) - [sin(a+1)cos(1)- cos(a+1)sin(1)]=

sin(a+1)cos(1)+cos(a+1)sin(1)- sin(a+1)cos(1)+ cos(a+1)sin(1)]

=>2[cos(a+1)sin(1)]

Answer 2

The identity for sinx - siny is 2 cos 1/2(x+y) sin 1/2 (x-y)
here, x = a+2 and y = a
x+y is 2a+2, times 1/2 is a+1
and x-y is a+2-a = 2 times 1/2 is 1

Answer 3

sin(a+2) = sin(a) cos(2) + cos(a) sin(2)
cos(2) = 1 - 2 sin²(1)
sin(2) = 2 sin(1) cos(1)

So

sin(a+2) - sin(a) = sin(a)[1 - 2 sin²(1)] + cos(a)[2 sin(1) cos(1)] - sin(a)
= sin(a) -2 sin²(1)sin(a) + 2 cos(a)sin(1) cos(1) - sin(a)
= 2 sin(1)[cos(a)cos(1) - sin(a)sin(1)]
= 2 sin(1)cos(a+1)