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

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We have

cos(n+1)xcos(n+2)x + sin(n+1)xsin(n+2)x

it is from the identity

cos(a-b) = cosa * cosb + sina * sinb

here a= (n+1)x  and b= (n+2)x

so,

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

= cos(nx+x-nx-2x)

= cos(-x)

= cosx { since cos(-x) = cosx }

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