Question

Prove that
sin(n+1)Asin(n-1)A + cos(n+1)Acos(n-1)A = cos2A

Answer 1

Let (n+1)A = x
Let (n-1)A = y
sinx*siny + cosx*cosy = cos (x - y)
(This is an identity you should know)
Now note that x -y = 2A , so

sinx*siny + cosx*cosy = cos (x - y)
= cos(2A)