Answer :

To Prove:  sin (n + 1)x  sin (n + 2)x + cos (n + 1)x  cos (n + 2)x = cos x

Proof:

R.H.S = cos x

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

We know by the formula that,

[Since, cos (A - B)=cos A cos B + sin A sin B ]

So let (n + 1) = A and (n + 2) = B from the L.H.S and putting in above formula,

sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos[(n + 1) x - (n + 2)x]               ......[ A = (n +1)x  and B = (n + 2)x ]

sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos(-x) = cos x

[As cos (- x) = cos x]

L.H.S= R.H.S

Hence, proved

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