Question

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

Answer 1

$L.H.S.=\sin (n+1)x\cdot \sin (n+2)x+\cos (n+1)x\cdot \cos (n+2)x$
Using formula $\cos (A-B)=\cos A\cos B+\sin A\sin B$
Let $A=(n+1)x$ and $B=(n+2)x$
Then $L.H.S.=\cos \left[\right(n+1)x-(n+2\left)x\right]$
$=\cos [nx+x-nx-2x]$
$=\cos (–x)$
$=\cos x=R.H.S.$
So, $L.H.S.=R.H.S.$
$\sin (n+1)x\cdot \sin (n+2)x+\cos (n+1)x\cdot \cos (n+2)x=\cos x$
Hence proved.