Question

URGENT

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

Answer 1

prove: sin(n+1)x sin(n+2)x+cos(n+1)x cos(n+2)x=cosx
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Identity: cos(s-t)
=cos(s)x*cos(t)x+sin(s)x*sin(t)x cos(s-t)
=cos(s)x*cos(t)x+sin(s)x*sin(t)x
cos((n+1)x)-((n+2)x=cos(n+1)x*cos(n+2)x+sin(n+1)x*sin(n+2)x
cos((nx+x)-(nx+2x))
=cos(n+1)x*cos(n+2)x+sin(n+1)x*sin(n+2)x
cos((nx+x)-(nx+2x))
=cos(nx+x-nx-2x)
=cos(-x)=cos(x)
verified: left side=right side 

Answer 2

Solution:

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

We take LHS

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

Let (n + 1)x = A

and (n + 2)x = B

So,

=> sin A sin B + cos A cos B

=> cos (A - B)

=> cos [(n - 1)x - (n - 2)x]

=> cos[nx + x - nx - 2n]

=> cos (-x)

=> cos x

So, LHS = RHS

Hence Proved!!