Prove that `cosx+^nC_1 cos2x+^nC_2 cos3x+....+^nC_n cos(n+1)x=2^n. cos^n x/2.cos((n+2)/2)x`

Prove that cos^2 x + cos^2 3x + cos^2 5x + ....+ to n terms= 1/2 [ n+ (sin4nx)/(2 sin2x)]

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

Prove that ""^nC_0+3""^nC_1+5""^nC_2+........+(2n+1)""^nC_n= (n+1)2^n .

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

Prove that ^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n

Prove that ""^nC_0+3*""^nC_1+5*""^nC_2+...+(2n+1)""^nC_n=(n+1)2^(n)

Prove that sum_ (k = 0) ^ (n) nC_ (k) sin Kx.cos (nK) x = 2 ^ (n-1) sin nx

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

Prove that Sigma_(K=0)^(n) ""^nC_(k) sin K x cos (n-K)x = 2^(n-1) sin x.

Prove that ""^nC_0+2*""^nC_1+3*""^nC_2+...+(n+1)""^nC_n=(n+2)2^(n-1)