# Prove the following: sin (n+1)x sin (n+2)x + cos (n+1)x cos (n+2)x = cos x

$\sin(n+1)x \sin(n+2)x+\cos(n+1)x \cos(n+2)x=\cos x\\*\therefore use\;the\; formula\;\sin A \sin B+\cos A \cos B=\cos(A-B)\\*Look\;at\;the\;LHS,\\* \Rightarrow \cos(n+1)x \cos(n+2)x+\sin(n+1)x \sin(n+2)x\\*\Rightarrow \cos((n+2)x-((n+1)x))=\cos(nx+2x-nx-x)=\cos x\\*Hence,\;proved$

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