Question

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

Answer 1

$\sin (n+1)x\sin (n+2)x={\sin }^2(n+1)\cos x+\sin (n+1)x\cos (n+1)x\sin x$

$\cos (n+1)x\cos (n+2)x={\cos }^2(n+1)x\cos x-\cos (n+1)x\sin (n+1)x\sin x$

$\sin (n+1)x\sin (n+2)x+\cos (n+1)x\cos (n+2)x=\cos x\left({\sin }^2\right(n+1)x+{\cos }^2(n+1\left)x\right)+\sin (n+1)x\cos (n+1)x\sin x-\cos (n+1)x\sin (n+1)x\sin x$

$=\cos x$