Simplify :
`Sin( n + 1 )x . Sin( n + 2 )x + Cos( n +1 )x . Cos( n + 2 )x`

sin (n + 1) A sin (n-1) A + cos (n + 1) A cos (n-1) A = cos2A

sin (n + 1) A * sin (n + 2) A + cos (n + 1) A * cos (n + 2) A = cos A

sin (n + 1) A * sin (n-1) A + cos (n + 1) A * cos (n-1) A = cos2A

cos (n + 1) A cos (n + 2) A + sin (n + 1) A sin (n + 2) A =

Prove that: sin(n+1)x sin(n+2)x+cos(n+1)x cos(n+2)x=c

sin(n+1)xsin(n+2)x+cos(n+1)xcos(n+2)x=cosx

(sin (n + 1) A + 2sin n + sin (n-1) A) / (cos (n-1) A-cos (n + 1) A) = (cot A) / (2)

Prove that nC_ (1) sin x * cos (n-1) x + nC_ (2) sin2x * cos (n-2) x + nC_ (3) sin3x * cos (n-3) x + ... + nC_ ( n) sin nx = 2 ^ (n-1) sin nn

sin x = 2 ^ (n) * cos (x) / (2) * cos (x) / (2 ^ (2)) * cos (x) / (2 ^ (3)) * ... * cos ( x) / (2 ^ (n)) * sin (x) / (2 ^ (n))