Question

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

Answer 1

To prove:- $\sin (n+1)x\sin (n+2)x+\cos (n+1)x\cos (n+2)x=\cos x$

Proof $:-$

$L.H.S.=$

$\sin (n+1)x\sin (n+2)x+\cos (n+1)x\cos (n+2)x$

$=\cos ((n+2)x-(n-1)x)\{\because \cos (A-B)=\sin A\sin B+\cos A\cos B\}$

$\Rightarrow =\cos \left((n+2-n-1)x\right)$

$\Rightarrow =\cos x=R.H.S.$

Hence proved.