Question

Prove that : $\sin \left(n+1\right)x\sin \left(n+2\right)x+\cos \left(n+1\right)x\cos \left(n+2\right)x=\cos x$

Answer 1

$LHS=\sin (n+1)x\sin (n+2)x+\cos (n+1)x\cos (n+2)x=\cos \left[\right(n+1)x-(n+2\left)x\right]=\cos [nx+x-nx-2x]=\cos x=RHS$