Amaths Proving of identities help2! D:

3. cos3x+sin3x=(1+2sin2x)(cosx-sinx)

cos3x+sin3x=(1+2sin2x)(cosx-sinx)
cos3x+sin3x
sin (A + B) = (sin A)(cos B) + (cos A)(sin B)
cos (A + B) = (cos A)cos B - (sin A)sin B
Let A=x and B=2x
sin (x + 2x) = (sin x)(cos 2x) + (cos x)(sin 2x)
cos (x +2x) = (cos x)cos 2x - (sin x)sin 2x
sin (3x) = (sin x)(cos 2x) + (cos x)(sin 2x)
cos (3x) = (cos x)cos 2x - (sin x)sin 2x
Adding both
LHS
=sin (3x) +cos (3x) =(sin x)(cos 2x) + (cos x)(sin 2x)+(cos x)cos 2x - (sin x)sin 2x
=(sin x)(cos 2x) + (cos x)(sin 2x)+(cos x)cos 2x - (sin x)sin 2x ---but cos (2x) = cos^2x - sin ^2 x
=(sin x)(cos^2x - sin ^2 x) +(cos x)(cos^2x - sin ^2 x) + (cos x)(sin 2x)- (sin x)sin 2x
={cos x-sin x)} {sin x (cos x +sin x) +cos x(cos x +sin x) }+ sin 2x {cos x-sin x)}
={cos x-sin x)} {sin x (cos x +sin x) +cos x(cos x +sin x) + sin 2x}
={cos x-sin x)} {sin x cos x +sin^2 x +cos^2 x +sin x cos x + sin 2x}
={cos x-sin x)} {1 +2 sin x cos x + sin 2x}
={cos x-sin x)} {1 +sin 2x + sin 2x}
={cos x-sin x)} {1 +2 sin 2x
=RHS}