Question

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

Step-by-step explanation:

WE KNOW THAT

$\cos(x) \cos(y) + \sin(x) \sin(y) = \cos(x - y)$

$\sin(n + 1)x . \sin(n + 2) x \: \: + \: \: \cos(n + 1) x \cos(n + 2 ) x \\ \: by \: \: rearrangement\\ = \cos(n + 1) x \cos(n + 2 ) x \: \: + \: \: \sin(n + 1)x . \sin(n + 2) x \\ \\ = \cos((n + 1)x - (n + 2)x) \\ \\ = \cos(nx + x - nx - 2x) \\ \\ = \cos( - x) \\ and \\ \: \: \cos( - x) = \cos(x)$

Proved