Prove that ((cos 8x − cos 7x)/(1 + 2 cos 5x)) = cos 3x − cos 2x


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Question Number 15555 by Tinkutara last updated on 11/Jun/17

$Prove that$ $\frac{\cos 8 x - \cos 7 x}{1 + 2 \cos 5 x} = \cos 3 x - \cos 2 x$
	Answered by ajfour last updated on 11/Jun/17

	$c o n s i d e r i n g$ $f (x) = (1 + 2 \cos 5 x) (\cos 3 x - \cos 2 x)$ $= (\cos 3 x - \cos 2 x) + 2 \cos 5 x \cos 3 x$ $- 2 \cos 5 x \cos 2 x$ $= (\cos 3 x - \cos 2 x) + (\cos 8 x + \cos 2 x)$ $- (\cos 7 x + \cos 3 x)$ $= \cos 8 x - \cos 7 x$ $t h e r e f o r e$ $\cos 3 x - \cos 2 x = \frac{\cos 8 x - \cos 7 x}{1 + 2 \cos 5 x} .$
	Commented by Tinkutara last updated on 13/Jun/17

	$Thanks Sir but I needed to prove LHS$ $= RHS . Can it be done ?$
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