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Prove that $\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+1) x$ $\cos (n+2) x=\cos x$
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L.H.S. $=\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x$
Write $A=(n+1) x$
$B=(n+2) x$
L.H.S. $=\sin A \sin B+\cos A \cos B$
$=\cos (A-B)$
$=\cos ((n+1) x-(n+2) x)$
$=\cos (x-2 x)$
$=\cos (-x) \quad[\because \cos (-\theta)=\cos \theta \forall \theta \in \mathrm{R}]$
$=\cos x=$ R.H.S.
Write $A=(n+1) x$
$B=(n+2) x$
L.H.S. $=\sin A \sin B+\cos A \cos B$
$=\cos (A-B)$
$=\cos ((n+1) x-(n+2) x)$
$=\cos (x-2 x)$
$=\cos (-x) \quad[\because \cos (-\theta)=\cos \theta \forall \theta \in \mathrm{R}]$
$=\cos x=$ R.H.S.
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