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


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


L.H.S. $=\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x$

$=\frac{1}{2}[2 \sin (n+1) x \sin (n+2) x+2 \cos (n+1) x \cos (n+2) x]$

$=\frac{1}{2}\left[\begin{array}{l}\cos \{(n+1) x-(n+2) x\}-\cos \{(n+1) x+(n+2) x\} \\ +\cos \{(n+1) x+(n+2) x\}+\cos \{(n+1) x-(n+2) x\}\end{array}\right]$

$\left[\begin{array}{l}\because-2 \sin A \sin B=\cos (A+B)-\cos (A-B) \\ 2 \cos A \cos B=\cos (A+B)+\cos (A-B)\end{array}\right]$

$=\frac{1}{2} \times 2 \cos \{(n+1) x-(n+2) x\}$

$=\cos (-x)=\cos x=$ R.H.S.

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