Question

PROVE THAT

sin ( n +1) X sin ( n+2) X + cos ( n +1) X cos ( n +2) X = cos X.

Answer 1

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


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Answer 2

we know
$\cos(x - y) = \sin(x) \sin(y) + \cos(x) \cos(y)$
so here
x=(n+2)X
y=(n+1)X
so compare and see the Lhs becomes
$\cos((n + 2)x - (n + 1)x) = \cos(x)$
.mark as brainliest if helped